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case pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
this : ∀ i ∈ s, f i * g i = 0
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
| have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
this✝ : ∀ i ∈ s, f i * g i = 0
this : ∑ i in s, f i * g i = ∑ i in s, 0
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
| simp [this] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : ¬((∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0)
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
| push_neg at H | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
| by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· | cases' H' with H' H' | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case pos.inl
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> | simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case pos.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> | simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤)
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
| replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case H'
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤)
⊢ (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤ | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· | simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H' | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
| have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq) | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
↑(∑ i in s, (fun i => ENNReal.toNNReal (f i)) i * (fun i => ENNReal.toNNReal (g i)) i) ≤
↑((∑ i in s, (fun i => ENNReal.toNNReal (f i)) i ^ p) ^ (1 / p) *
(∑ i in s, (fun i => ENNReal.toNNReal (g i)) i ^ q) ^ (1 / q))
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
| simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
| convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
| convert this using 1 <;> [skip; congr 2] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
| convert this using 1 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, f i * g i = ∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [ | skip | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [ | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_4
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) =
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹ | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; | congr 2 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, f i * g i = ∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [ | skip | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [ | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, f i ^ p = ∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; | skip | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ 1 / p = p⁻¹ | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; | simp | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, g i ^ q = ∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; | skip | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ 1 / q = q⁻¹ | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; | simp | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, f i * g i = ∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
i : ι
hi : i ∈ s
⊢ f i * g i = ↑(ENNReal.toNNReal (f i)) * ↑(ENNReal.toNNReal (g i)) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, f i ^ p = ∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
i : ι
hi : i ∈ s
⊢ f i ^ p = ↑(ENNReal.toNNReal (f i)) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
⊢ ∑ i in s, g i ^ q = ∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ p⁻¹ * (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ q) ^ q⁻¹
i : ι
hi : i ∈ s
⊢ g i ^ q = ↑(ENNReal.toNNReal (g i)) ^ q | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
| cases' eq_or_lt_of_le hp with hp hp | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
| Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
case inl
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp✝ : 1 ≤ p
hp : 1 = p
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· | simp [← hp] | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
| let q : ℝ := p / (p - 1) | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
| Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
| have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp] | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
| Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
⊢ Real.IsConjugateExponent p q | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by | rw [Real.isConjugateExponent_iff hp] | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
| have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
| Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
hp₁ : 1 / p * p = 1
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
| have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
| Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
hp₁ : 1 / p * p = 1
⊢ 1 / q * p = p - 1 | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
| rw [← hpq.div_conj_eq_sub_one] | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
| Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
hp₁ : 1 / p * p = 1
⊢ 1 / q * p = p / q | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
| ring | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
| Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
hp₁ : 1 / p * p = 1
hq : 1 / q * p = p - 1
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
| simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
| Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
| by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· | cases' H' with H' H' | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case pos.inl
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> | simp [H', -one_div] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case pos.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> | simp [H', -one_div] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤)
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
| have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤)
pos : 0 < p
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
| replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case H'
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤)
pos : 0 < p
⊢ (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤ | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· | simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H' | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
| have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp) | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
↑((∑ i in s, ((fun i => ENNReal.toNNReal (f i)) i + (fun i => ENNReal.toNNReal (g i)) i) ^ p) ^ (1 / p)) ≤
↑((∑ i in s, (fun i => ENNReal.toNNReal (f i)) i ^ p) ^ (1 / p) +
(∑ i in s, (fun i => ENNReal.toNNReal (g i)) i ^ p) ^ (1 / p))
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
| push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
| convert this using 2 <;> [skip; congr 1; congr 1] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
| convert this using 2 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case h.e'_3.h.e'_5
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
⊢ ∑ i in s, (f i + g i) ^ p = ∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [ | skip | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [ | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_5
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
⊢ (∑ i in s, f i ^ p) ^ (1 / p) = (∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; | congr 1 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_6
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
⊢ (∑ i in s, g i ^ p) ^ (1 / p) = (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; | congr 1 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case h.e'_3.h.e'_5
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
⊢ ∑ i in s, (f i + g i) ^ p = ∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case h.e'_3.h.e'_5
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
i : ι
hi : i ∈ s
⊢ (f i + g i) ^ p = (↑(ENNReal.toNNReal (f i)) + ↑(ENNReal.toNNReal (g i))) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| simp [H'.1 i hi, H'.2 i hi] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_5.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
⊢ ∑ i in s, f i ^ p = ∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_5.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
i : ι
hi : i ∈ s
⊢ f i ^ p = ↑(ENNReal.toNNReal (f i)) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| simp [H'.1 i hi, H'.2 i hi] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_6.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
⊢ ∑ i in s, g i ^ p = ∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_6.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)
i : ι
hi : i ∈ s
⊢ g i ^ p = ↑(ENNReal.toNNReal (g i)) ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjugateExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
/-!
# Mean value inequalities
In this file we prove several inequalities for finite sums, including AM-GM inequality,
Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of
these inequalities are available in `MeasureTheory.MeanInequalities`.
## Main theorems
### AM-GM inequality:
The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
are two non-negative vectors and $\sum_{i\in s} w_i=1$, then
$$
\prod_{i\in s} z_i^{w_i} ≤ \sum_{i\in s} w_iz_i.
$$
The classical version is a special case of this inequality for $w_i=\frac{1}{n}$.
We prove a few versions of this inequality. Each of the following lemmas comes in two versions:
a version for real-valued non-negative functions is in the `Real` namespace, and a version for
`NNReal`-valued functions is in the `NNReal` namespace.
- `geom_mean_le_arith_mean_weighted` : weighted version for functions on `Finset`s;
- `geom_mean_le_arith_mean2_weighted` : weighted version for two numbers;
- `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers;
- `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers.
### Young's inequality
Young's inequality says that for non-negative numbers `a`, `b`, `p`, `q` such that
$\frac{1}{p}+\frac{1}{q}=1$ we have
$$
ab ≤ \frac{a^p}{p} + \frac{b^q}{q}.
$$
This inequality is a special case of the AM-GM inequality. It is then used to prove Hölder's
inequality (see below).
### Hölder's inequality
The inequality says that for two conjugate exponents `p` and `q` (i.e., for two positive numbers
such that $\frac{1}{p}+\frac{1}{q}=1$) and any two non-negative vectors their inner product is
less than or equal to the product of the $L_p$ norm of the first vector and the $L_q$ norm of the
second vector:
$$
\sum_{i\in s} a_ib_i ≤ \sqrt[p]{\sum_{i\in s} a_i^p}\sqrt[q]{\sum_{i\in s} b_i^q}.
$$
We give versions of this result in `ℝ`, `ℝ≥0` and `ℝ≥0∞`.
There are at least two short proofs of this inequality. In our proof we prenormalize both vectors,
then apply Young's inequality to each $a_ib_i$. Another possible proof would be to deduce this
inequality from the generalized mean inequality for well-chosen vectors and weights.
### Minkowski's inequality
The inequality says that for `p ≥ 1` the function
$$
\|a\|_p=\sqrt[p]{\sum_{i\in s} a_i^p}
$$
satisfies the triangle inequality $\|a+b\|_p\le \|a\|_p+\|b\|_p$.
We give versions of this result in `Real`, `ℝ≥0` and `ℝ≥0∞`.
We deduce this inequality from Hölder's inequality. Namely, Hölder inequality implies that $\|a\|_p$
is the maximum of the inner product $\sum_{i\in s}a_ib_i$ over `b` such that $\|b\|_q\le 1$. Now
Minkowski's inequality follows from the fact that the maximum value of the sum of two functions is
less than or equal to the sum of the maximum values of the summands.
## TODO
- each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them
is to define `StrictConvexOn` functions.
- generalized mean inequality with any `p ≤ q`, including negative numbers;
- prove that the power mean tends to the geometric mean as the exponent tends to zero.
-/
universe u v
open Finset Classical BigOperators NNReal ENNReal
set_option linter.uppercaseLean3 false
noncomputable section
variable {ι : Type u} (s : Finset ι)
section GeomMeanLEArithMean
/-! ### AM-GM inequality -/
namespace Real
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i in s, z i ^ w i ≤ ∑ i in s, w i * z i := by
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
-- for `exp` and numbers `log (z i)` with weights `w i`.
· simp only [not_exists, not_and, Ne.def, Classical.not_not] at A
have := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)
simp only [exp_sum, (· ∘ ·), smul_eq_mul, mul_comm (w _) (log _)] at this
convert this using 1 <;> [apply prod_congr rfl;apply sum_congr rfl] <;> intro i hi
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· exact rpow_def_of_pos hz _
· cases' eq_or_lt_of_le (hz i hi) with hz hz
· simp [A i hi hz.symm]
· rw [exp_log hz]
#align real.geom_mean_le_arith_mean_weighted Real.geom_mean_le_arith_mean_weighted
/-- AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**. --/
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ ≤ (∑ i in s, w i * z i) / (∑ i in s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i in s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg_of_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_assoc, ← Finset.sum_mul, mul_comm]
· exact fun _ hi => div_nonneg (hw _ hi) (le_of_lt hw')
· simp_rw [div_eq_mul_inv, ← Finset.sum_mul]
exact mul_inv_cancel (by linarith)
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = x :=
calc
∏ i in s, z i ^ w i = ∏ i in s, x ^ w i := by
refine' prod_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with h₀ | h₀
· rw [h₀, rpow_zero, rpow_zero]
· rw [hx i hi h₀]
_ = x := by
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
have : (∑ i in s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero
obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
rw [← hx i his hi]
exact hz i his
#align real.geom_mean_weighted_of_constant Real.geom_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i in s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i in s, w i * z i = x :=
calc
∑ i in s, w i * z i = ∑ i in s, w i * x := by
refine' sum_congr rfl fun i hi => _
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
#align real.arith_mean_weighted_of_constant Real.arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i in s, z i ^ w i = ∑ i in s, w i * z i := by
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
#align real.geom_mean_eq_arith_mean_weighted_of_constant Real.geom_mean_eq_arith_mean_weighted_of_constant
end Real
namespace NNReal
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for `NNReal`-valued functions. -/
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, z i ^ (w i : ℝ)) ≤ ∑ i in s, w i * z i :=
mod_cast
Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)
(by assumption_mod_cast) fun i _ => (z i).coe_nonneg
#align nnreal.geom_mean_le_arith_mean_weighted NNReal.geom_mean_le_arith_mean_weighted
/-- The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `NNReal` numbers. -/
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using
geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]
#align nnreal.geom_mean_le_arith_mean2_weighted NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃]
#align nnreal.geom_mean_le_arith_mean3_weighted NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 →
p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ := by
simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, ← add_assoc,
mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄]
#align nnreal.geom_mean_le_arith_mean4_weighted NNReal.geom_mean_le_arith_mean4_weighted
end NNReal
namespace Real
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ :=
NNReal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean2_weighted Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ :=
NNReal.geom_mean_le_arith_mean3_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩
⟨p₃, hp₃⟩ <|
NNReal.coe_eq.1 hw
#align real.geom_mean_le_arith_mean3_weighted Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ :=
NNReal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩ ⟨p₁, hp₁⟩
⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ <|
NNReal.coe_eq.1 <| by assumption
#align real.geom_mean_le_arith_mean4_weighted Real.geom_mean_le_arith_mean4_weighted
end Real
end GeomMeanLEArithMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- Young's inequality, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.IsConjugateExponent q) : a * b ≤ a ^ p / p + b ^ q / q := by
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj
#align real.young_inequality_of_nonneg Real.young_inequality_of_nonneg
/-- Young's inequality, a version for arbitrary real numbers. -/
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ |a| ^ p / p + |b| ^ q / q :=
calc
a * b ≤ |a * b| := le_abs_self (a * b)
_ = |a| * |b| := (abs_mul a b)
_ ≤ |a| ^ p / p + |b| ^ q / q :=
Real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq
#align real.young_inequality Real.young_inequality
end Real
namespace NNReal
/-- Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, NNReal.coe_eq.2 hpq⟩
#align nnreal.young_inequality NNReal.young_inequality
/-- Young's inequality, `ℝ≥0` version with real conjugate exponents. -/
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q := by
nth_rw 1 [← Real.coe_toNNReal p hpq.nonneg]
nth_rw 1 [← Real.coe_toNNReal q hpq.symm.nonneg]
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal
#align nnreal.young_inequality_real NNReal.young_inequality_real
end NNReal
namespace ENNReal
/-- Young's inequality, `ℝ≥0∞` version with real conjugate exponents. -/
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q := by
by_cases h : a = ⊤ ∨ b = ⊤
· refine' le_trans le_top (le_of_eq _)
repeat rw [div_eq_mul_inv]
cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
push_neg at h
-- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
@coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
@coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
#align ennreal.young_inequality ENNReal.young_inequality
end ENNReal
end Young
section HolderMinkowski
/-! ### Hölder's and Minkowski's inequalities -/
namespace NNReal
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p ≤ 1) (hg : ∑ i in s, g i ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 := by
have hp_ne_zero : Real.toNNReal p ≠ 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm
have hq_ne_zero : Real.toNNReal q ≠ 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm
calc
∑ i in s, f i * g i ≤ ∑ i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=
Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq
_ = (∑ i in s, f i ^ p) / Real.toNNReal p + (∑ i in s, g i ^ q) / Real.toNNReal q := by
rw [sum_add_distrib, sum_div, sum_div]
_ ≤ 1 / Real.toNNReal p + 1 / Real.toNNReal q := by
refine' add_le_add _ _
· rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]
· rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]
_ = 1 := hpq.inv_add_inv_conj_nnreal
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : ∑ i in s, f i ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul,
inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]
intro i his
left
rw [sum_eq_zero_iff] at hf
exact (rpow_eq_zero_iff.mp (hf i his)).left
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. -/
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases hF_zero : ∑ i in s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i in s, g i ^ q = 0
· calc
∑ i in s, f i * g i = ∑ i in s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i in s, g i ^ q) ^ (1 / q) * (∑ i in s, f i ^ p) ^ (1 / p) :=
(inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)
_ = (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := mul_comm _ _
let f' i := f i / (∑ i in s, f i ^ p) ^ (1 / p)
let g' i := g i / (∑ i in s, g i ^ q) ^ (1 / q)
suffices (∑ i in s, f' i * g' i) ≤ 1 by
simp_rw [div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff, one_mul] at this
refine' mul_ne_zero _ _
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hF_zero
· rw [Ne.def, rpow_eq_zero_iff, not_and_or]
exact Or.inl hG_zero
refine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one,
div_self hF_zero]
· simp_rw [div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero,
rpow_one, div_self hG_zero]
#align nnreal.inner_le_Lp_mul_Lq NNReal.inner_le_Lp_mul_Lq
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_hasSum`. -/
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
have H₁ : ∀ s : Finset ι,
∑ i in s, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
intro s
refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hf
· rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]
exact sum_le_tsum _ (fun _ _ => zero_le _) hg
have bdd : BddAbove (Set.range fun s => ∑ i in s, f i * g i) := by
refine' ⟨(∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q), _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
exact ⟨H₂, tsum_le_of_sum_le H₂ H₁⟩
#align nnreal.inner_le_Lp_mul_Lq_tsum NNReal.inner_le_Lp_mul_Lq_tsum
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).1
#align nnreal.summable_mul_of_Lp_Lq NNReal.summable_mul_of_Lp_Lq
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjugateExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum hpq hf hg).2
#align nnreal.inner_le_Lp_mul_Lq_tsum' NNReal.inner_le_Lp_mul_Lq_tsum'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.IsConjugateExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
(hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
refine' ⟨∑' i, f i * g i, _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum
#align nnreal.inner_le_Lp_mul_Lq_has_sum NNReal.inner_le_Lp_mul_Lq_hasSum
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [NNReal.mul_rpow, ← NNReal.rpow_mul, hp₁, hq, one_mul, one_rpow, rpow_one,
Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align nnreal.rpow_sum_le_const_mul_sum_rpow NNReal.rpow_sum_le_const_mul_sum_rpow
/-- The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. -/
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjugateExponent q) :
IsGreatest ((fun g : ι → ℝ≥0 => ∑ i in s, f i * g i) '' { g | ∑ i in s, g i ^ q ≤ 1 })
((∑ i in s, f i ^ p) ^ (1 / p)) := by
constructor
· use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)
by_cases hf : ∑ i in s, f i ^ p = 0
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine' fun y => mul_div_cancel_left_of_imp fun h => _
simp [h, hpq.ne_zero]
simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←
rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]
rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]
simpa [hpq.symm.ne_zero] using hf
· rintro _ ⟨g, hg, rfl⟩
apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
simpa only [mul_one] using
mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _
#align nnreal.is_greatest_Lp NNReal.isGreatest_Lp
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `NNReal`-valued functions. -/
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with (rfl | hp);
· simp [Finset.sum_add_distrib]
have hpq := Real.isConjugateExponent_conjugateExponent hp
have := isGreatest_Lp s (f + g) hpq
simp only [Pi.add_apply, add_mul, sum_add_distrib] at this
rcases this.1 with ⟨φ, hφ, H⟩
rw [← H]
exact
add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)
#align nnreal.Lp_add_le NNReal.Lp_add_le
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have H₁ : ∀ s : Finset ι,
(∑ i in s, (f i + g i) ^ p) ≤
((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [← NNReal.rpow_one_div_le_iff pos]
refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>
rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>
refine' sum_le_tsum _ (fun _ _ => zero_le _) _
exacts [hf, hg]
have bdd : BddAbove (Set.range fun s => ∑ i in s, (f i + g i) ^ p) := by
refine' ⟨((∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p)) ^ p, _⟩
rintro a ⟨s, rfl⟩
exact H₁ s
have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
refine' ⟨H₂, _⟩
rw [NNReal.rpow_one_div_le_iff pos]
refine' tsum_le_of_sum_le H₂ H₁
#align nnreal.Lp_add_le_tsum NNReal.Lp_add_le_tsum
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum hp hf hg).1
#align nnreal.summable_Lp_add NNReal.summable_Lp_add
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum hp hf hg).2
#align nnreal.Lp_add_le_tsum' NNReal.Lp_add_le_tsum'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine' ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), _, _⟩
· simpa [hA, hB] using H₂
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum
#align nnreal.Lp_add_le_has_sum NNReal.Lp_add_le_hasSum
end NNReal
namespace Real
variable (f g : ι → ℝ) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : IsConjugateExponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) := by
have :=
NNReal.coe_le_coe.2
(NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)
hpq)
push_cast at this
refine' le_trans (sum_le_sum fun i _ => _) this
simp only [← abs_mul, le_abs_self]
#align real.inner_le_Lp_mul_Lq Real.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. -/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, |f i| ^ p := by
have :=
NNReal.coe_le_coe.2
(NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
push_cast at this
exact this
#align real.rpow_sum_le_const_mul_sum_rpow Real.rpow_sum_le_const_mul_sum_rpow
-- for some reason `exact_mod_cast` can't replace this argument
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `Real`-valued functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) := by
have :=
NNReal.coe_le_coe.2
(NNReal.Lp_add_le s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩) hp)
push_cast at this
refine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this <;>
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpow]
#align real.Lp_add_le Real.Lp_add_le
variable {f g}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjugateExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
(hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
convert inner_le_Lp_mul_Lq s f g hpq using 3 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi]
#align real.inner_le_Lp_mul_Lq_of_nonneg Real.inner_le_Lp_mul_Lq_of_nonneg
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_hasSum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
Summable fun i => f i * g i :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1
#align real.summable_mul_of_Lp_Lq_of_nonneg Real.summable_mul_of_Lp_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjugateExponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) :=
(inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2
#align real.inner_le_Lp_mul_Lq_tsum_of_nonneg' Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjugateExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine' ⟨C, C.prop, hC, _⟩
norm_cast
#align real.inner_le_Lp_mul_Lq_has_sum_of_nonneg Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. -/
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i in s, f i ^ p := by
convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
simp only [abs_of_nonneg, hf i hi]
#align real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. -/
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
convert Lp_add_le s f g hp using 2 <;> [skip;congr 1;congr 1] <;> apply sum_congr rfl <;>
intro i hi <;>
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]
#align real.Lp_add_le_of_nonneg Real.Lp_add_le_of_nonneg
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th powers, see `Lp_add_le_hasSum_of_nonneg`. -/
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
#align real.Lp_add_le_tsum_of_nonneg Real.Lp_add_le_tsum_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
Summable fun i => (f i + g i) ^ p :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1
#align real.summable_Lp_add_of_nonneg Real.summable_Lp_add_of_nonneg
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) :=
(Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2
#align real.Lp_add_le_tsum_of_nonneg' Real.Lp_add_le_tsum_of_nonneg'
/-- Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
(hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at hfA hgB
norm_cast at hfA hgB
obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
use C
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce
norm_cast
exact ⟨zero_le _, hC₁, hC₂⟩
#align real.Lp_add_le_has_sum_of_nonneg Real.Lp_add_le_hasSum_of_nonneg
end Real
namespace ENNReal
variable (f g : ι → ℝ≥0∞) {p q : ℝ}
/-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) := by
by_cases H : (∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
· simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]
have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this
simp [this]
push_neg at H
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq)
simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]
#align ennreal.inner_le_Lp_mul_Lq ENNReal.inner_le_Lp_mul_Lq
/-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ^ p := by
cases' eq_or_lt_of_le hp with hp hp
· simp [← hp]
let q : ℝ := p / (p - 1)
have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring
simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg
#align ennreal.rpow_sum_le_const_mul_sum_rpow ENNReal.rpow_sum_le_const_mul_sum_rpow
/-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| simp [H'.1 i hi, H'.2 i hi] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) := by
by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
· cases' H' with H' H' <;> simp [H', -one_div]
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
· simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'
have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp)
push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
convert this using 2 <;> [skip; congr 1; congr 1] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
| Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | Mathlib_Analysis_MeanInequalities |
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
⊢ p / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
| obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
r : R
hr : p = GCDMonoid.gcd p q * r
⊢ p / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
| obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp) | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
r : R
hr : p = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ p / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
| nth_rw 1 [hr] | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
r : R
hr : p = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
| rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0] | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
r : R
hr : p = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ r ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
| exact r0 | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
⊢ q / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
exact r0
#align left_div_gcd_ne_zero left_div_gcd_ne_zero
theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
| obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
⊢ q / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
exact r0
#align left_div_gcd_ne_zero left_div_gcd_ne_zero
theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
| obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq) | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ q / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
exact r0
#align left_div_gcd_ne_zero left_div_gcd_ne_zero
theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
| nth_rw 1 [hr] | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
exact r0
#align left_div_gcd_ne_zero left_div_gcd_ne_zero
theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
nth_rw 1 [hr]
| rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0] | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
nth_rw 1 [hr]
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ r ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
exact r0
#align left_div_gcd_ne_zero left_div_gcd_ne_zero
theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
| exact r0 | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
R : Type ?u.6733
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
a b : R
⊢ Associated (gcd a b * lcm a b) (a * b) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
/-!
# Lemmas about Euclidean domains
Various about Euclidean domains are proved; all of them seem to be true
more generally for principal ideal domains, so these lemmas should
probably be reproved in more generality and this file perhaps removed?
## Tags
euclidean domain
-/
section
open EuclideanDomain Set Ideal
section GCDMonoid
variable {R : Type*} [EuclideanDomain R] [GCDMonoid R] {p q : R}
theorem gcd_ne_zero_of_left (hp : p ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q)
#align gcd_ne_zero_of_left gcd_ne_zero_of_left
theorem gcd_ne_zero_of_right (hp : q ≠ 0) : GCDMonoid.gcd p q ≠ 0 := fun h =>
hp <| eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q)
#align gcd_ne_zero_of_right gcd_ne_zero_of_right
theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
exact r0
#align left_div_gcd_ne_zero left_div_gcd_ne_zero
theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
exact r0
#align right_div_gcd_ne_zero right_div_gcd_ne_zero
theorem isCoprime_div_gcd_div_gcd (hq : q ≠ 0) :
IsCoprime (p / GCDMonoid.gcd p q) (q / GCDMonoid.gcd p q) :=
(gcd_isUnit_iff _ _).1 <|
isUnit_gcd_of_eq_mul_gcd
(EuclideanDomain.mul_div_cancel' (gcd_ne_zero_of_right hq) <| gcd_dvd_left _ _).symm
(EuclideanDomain.mul_div_cancel' (gcd_ne_zero_of_right hq) <| gcd_dvd_right _ _).symm <|
gcd_ne_zero_of_right hq
#align is_coprime_div_gcd_div_gcd isCoprime_div_gcd_div_gcd
end GCDMonoid
namespace EuclideanDomain
/-- Create a `GCDMonoid` whose `GCDMonoid.gcd` matches `EuclideanDomain.gcd`. -/
-- porting note: added `DecidableEq R`
def gcdMonoid (R) [EuclideanDomain R] [DecidableEq R] : GCDMonoid R where
gcd := gcd
lcm := lcm
gcd_dvd_left := gcd_dvd_left
gcd_dvd_right := gcd_dvd_right
dvd_gcd := dvd_gcd
gcd_mul_lcm a b := by | rw [EuclideanDomain.gcd_mul_lcm] | /-- Create a `GCDMonoid` whose `GCDMonoid.gcd` matches `EuclideanDomain.gcd`. -/
-- porting note: added `DecidableEq R`
def gcdMonoid (R) [EuclideanDomain R] [DecidableEq R] : GCDMonoid R where
gcd := gcd
lcm := lcm
gcd_dvd_left := gcd_dvd_left
gcd_dvd_right := gcd_dvd_right
dvd_gcd := dvd_gcd
gcd_mul_lcm a b := by | Mathlib.RingTheory.EuclideanDomain.71_0.j84WZGwHDjQhSAS | /-- Create a `GCDMonoid` whose `GCDMonoid.gcd` matches `EuclideanDomain.gcd`. -/
-- porting note: added `DecidableEq R`
def gcdMonoid (R) [EuclideanDomain R] [DecidableEq R] : GCDMonoid R where
gcd | Mathlib_RingTheory_EuclideanDomain |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
⊢ (aeval f) (charpoly f) = 0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
| apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1 | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
| Mathlib.LinearAlgebra.Charpoly.Basic.64_0.6NA9VnT03sJgAKk | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
⊢ (AlgEquiv.toLinearEquiv (algEquivMatrix (chooseBasis R M))) ((aeval f) (charpoly f)) = 0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
| rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def] | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
| Mathlib.LinearAlgebra.Charpoly.Basic.64_0.6NA9VnT03sJgAKk | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
⊢ (aeval (↑(algEquivMatrix (chooseBasis R M)) f)) (Matrix.charpoly ((toMatrix (chooseBasis R M) (chooseBasis R M)) f)) =
0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
| exact Matrix.aeval_self_charpoly _ | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
| Mathlib.LinearAlgebra.Charpoly.Basic.64_0.6NA9VnT03sJgAKk | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
k : ℕ
⊢ f ^ k = (aeval f) (X ^ k %ₘ charpoly f) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
| rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X] | /-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
| Mathlib.LinearAlgebra.Charpoly.Basic.90_0.6NA9VnT03sJgAKk | /-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
⊢ coeff (minpoly R f) 0 ≠ 0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
| intro h | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
| obtain ⟨P, hP⟩ := X_dvd_iff.2 h | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
| have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
⊢ degree P < degree (minpoly R f) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
| rw [hP, mul_comm] | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
⊢ degree P < degree (P * X) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
| refine' degree_lt_degree_mul_X fun h => _ | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h✝ : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
h : P = 0
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
| rw [h, mul_zero] at hP | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h✝ : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = 0
h : P = 0
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
| exact minpoly.ne_zero (isIntegral f) hP | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
| have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
exact minpoly.monic (isIntegral f) | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
⊢ Monic P | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
| suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
this : Monic (minpoly R f)
⊢ Monic P | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
| rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
⊢ Monic (minpoly R f) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
| exact minpoly.monic (isIntegral f) | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
hPmonic : Monic P
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
exact minpoly.monic (isIntegral f)
| have hzero : aeval f (minpoly R f) = 0 := minpoly.aeval _ _ | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
exact minpoly.monic (isIntegral f)
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
hPmonic : Monic P
hzero : (aeval f) (minpoly R f) = 0
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
exact minpoly.monic (isIntegral f)
have hzero : aeval f (minpoly R f) = 0 := minpoly.aeval _ _
| simp only [hP, mul_eq_comp, ext_iff, hf, aeval_X, map_eq_zero_iff, coe_comp, AlgHom.map_mul,
zero_apply, Function.comp_apply] at hzero | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
exact minpoly.monic (isIntegral f)
have hzero : aeval f (minpoly R f) = 0 := minpoly.aeval _ _
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
hPmonic : Monic P
hzero : ∀ (x : M), ((aeval f) P) x = 0
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.charpoly.basic from "leanprover-community/mathlib"@"d3e8e0a0237c10c2627bf52c246b15ff8e7df4c0"
/-!
# Characteristic polynomial
We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and
free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f`
in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.
## Main definition
* `LinearMap.charpoly f` : the characteristic polynomial of `f : M →ₗ[R] M`.
-/
universe u v w
variable {R : Type u} {M : Type v} [CommRing R] [Nontrivial R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
open Matrix Polynomial
noncomputable section
open Module.Free Polynomial Matrix
namespace LinearMap
section Basic
/-- The characteristic polynomial of `f : M →ₗ[R] M`. -/
def charpoly : R[X] :=
(toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly
#align linear_map.charpoly LinearMap.charpoly
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly :=
rfl
#align linear_map.charpoly_def LinearMap.charpoly_def
end Basic
section Coeff
theorem charpoly_monic : f.charpoly.Monic :=
Matrix.charpoly_monic _
#align linear_map.charpoly_monic LinearMap.charpoly_monic
end Coeff
section CayleyHamilton
/-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1
rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def]
exact Matrix.aeval_self_charpoly _
#align linear_map.aeval_self_charpoly LinearMap.aeval_self_charpoly
theorem isIntegral : IsIntegral R f :=
⟨f.charpoly, ⟨charpoly_monic f, aeval_self_charpoly f⟩⟩
#align linear_map.is_integral LinearMap.isIntegral
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
[FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly f)
#align linear_map.minpoly_dvd_charpoly LinearMap.minpoly_dvd_charpoly
/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
(aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm
#align linear_map.aeval_eq_aeval_mod_charpoly LinearMap.aeval_eq_aeval_mod_charpoly
/-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
#align linear_map.pow_eq_aeval_mod_charpoly LinearMap.pow_eq_aeval_mod_charpoly
variable {f}
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
exact minpoly.monic (isIntegral f)
have hzero : aeval f (minpoly R f) = 0 := minpoly.aeval _ _
simp only [hP, mul_eq_comp, ext_iff, hf, aeval_X, map_eq_zero_iff, coe_comp, AlgHom.map_mul,
zero_apply, Function.comp_apply] at hzero
| exact not_le.2 hdegP (minpoly.min _ _ hPmonic (ext hzero)) | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
exact minpoly.monic (isIntegral f)
have hzero : aeval f (minpoly R f) = 0 := minpoly.aeval _ _
simp only [hP, mul_eq_comp, ext_iff, hf, aeval_X, map_eq_zero_iff, coe_comp, AlgHom.map_mul,
zero_apply, Function.comp_apply] at hzero
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
α : Type u
β : Type u_1
w x✝ y✝ z : α
inst✝ : GeneralizedBooleanAlgebra α
x y : α
⊢ x \ y ⊔ x ⊓ y = x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by | rw [sup_comm, sup_inf_sdiff] | @[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by | Mathlib.Order.BooleanAlgebra.106_0.ewE75DLNneOU8G5 | @[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x✝ y✝ z : α
inst✝ : GeneralizedBooleanAlgebra α
x y : α
⊢ x \ y ⊓ (x ⊓ y) = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by | rw [inf_comm, inf_inf_sdiff] | @[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by | Mathlib.Order.BooleanAlgebra.110_0.ewE75DLNneOU8G5 | @[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
src✝ : Bot α := toBot
a : α
⊢ ⊥ ≤ a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
| rw [← inf_inf_sdiff a a, inf_assoc] | instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
| Mathlib.Order.BooleanAlgebra.115_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
src✝ : Bot α := toBot
a : α
⊢ a ⊓ (a ⊓ a \ a) ≤ a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
| exact inf_le_left | instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
| Mathlib.Order.BooleanAlgebra.115_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x
i : x ⊓ y ⊓ z = ⊥
⊢ x \ y = z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
| conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
| Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x
i : x ⊓ y ⊓ z = ⊥
| x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => | rw [← sup_inf_sdiff x y, sup_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x
i : x ⊓ y ⊓ z = ⊥
| x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => | rw [← sup_inf_sdiff x y, sup_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x
i : x ⊓ y ⊓ z = ⊥
| x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => | rw [← sup_inf_sdiff x y, sup_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
⊢ x \ y = z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
| rw [sup_comm] at s | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
| Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
⊢ x \ y = z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
| conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
| Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
| ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => | rw [← inf_inf_sdiff x y, inf_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
| ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => | rw [← inf_inf_sdiff x y, inf_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
| ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => | rw [← inf_inf_sdiff x y, inf_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = x \ y ⊓ (x ⊓ y)
⊢ x \ y = z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras
generalize the (classical) logic of propositions and the lattice of subsets of a set.
Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which
do not necessarily have a top element (`⊤`) (and hence not all elements may have complements). One
example in mathlib is `Finset α`, the type of all finite subsets of an arbitrary
(not-necessarily-finite) type `α`.
`GeneralizedBooleanAlgebra α` is defined to be a distributive lattice with bottom (`⊥`) admitting
a *relative* complement operator, written using "set difference" notation as `x \ y` (`sdiff x y`).
For convenience, the `BooleanAlgebra` type class is defined to extend `GeneralizedBooleanAlgebra`
so that it is also bundled with a `\` operator.
(A terminological point: `x \ y` is the complement of `y` relative to the interval `[⊥, x]`. We do
not yet have relative complements for arbitrary intervals, as we do not even have lattice
intervals.)
## Main declarations
* `GeneralizedBooleanAlgebra`: a type class for generalized Boolean algebras
* `BooleanAlgebra`: a type class for Boolean algebras.
* `Prop.booleanAlgebra`: the Boolean algebra instance on `Prop`
## Implementation notes
The `sup_inf_sdiff` and `inf_inf_sdiff` axioms for the relative complement operator in
`GeneralizedBooleanAlgebra` are taken from
[Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations).
[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative
complement operator `a \ b` for all `a`, `b`. Instead, the postulates there amount to an assumption
that for all `a, b : α` where `a ≤ b`, the equations `x ⊔ a = b` and `x ⊓ a = ⊥` have a solution
`x`. `Disjoint.sdiff_unique` proves that this `x` is in fact `b \ a`.
## References
* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>
* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
## Tags
generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl
-/
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
/-!
### Generalized Boolean algebras
Some of the lemmas in this section are from:
* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]
* <https://ncatlab.org/nlab/show/relative+complement>
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
-/
/-- A generalized Boolean algebra is a distributive lattice with `⊥` and a relative complement
operation `\` (called `sdiff`, after "set difference") satisfying `(a ⊓ b) ⊔ (a \ b) = a` and
`(a ⊓ b) ⊓ (a \ b) = ⊥`, i.e. `a \ b` is the complement of `b` in `a`.
This is a generalization of Boolean algebras which applies to `Finset α` for arbitrary
(not-necessarily-`Fintype`) `α`. -/
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, SDiff α, Bot α where
/-- For any `a`, `b`, `(a ⊓ b) ⊔ (a / b) = a` -/
sup_inf_sdiff : ∀ a b : α, a ⊓ b ⊔ a \ b = a
/-- For any `a`, `b`, `(a ⊓ b) ⊓ (a / b) = ⊥` -/
inf_inf_sdiff : ∀ a b : α, a ⊓ b ⊓ a \ b = ⊥
#align generalized_boolean_algebra GeneralizedBooleanAlgebra
-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,
-- however we'd need another type class for lattices with bot, and all the API for that.
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
@[simp]
theorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \ y = x :=
GeneralizedBooleanAlgebra.sup_inf_sdiff _ _
#align sup_inf_sdiff sup_inf_sdiff
@[simp]
theorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \ y = ⊥ :=
GeneralizedBooleanAlgebra.inf_inf_sdiff _ _
#align inf_inf_sdiff inf_inf_sdiff
@[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]
#align sup_sdiff_inf sup_sdiff_inf
@[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]
#align inf_sdiff_inf inf_sdiff_inf
-- see Note [lower instance priority]
instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
exact inf_le_left }
#align generalized_boolean_algebra.to_order_bot GeneralizedBooleanAlgebra.toOrderBot
theorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \ y) :=
disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le
#align disjoint_inf_sdiff disjoint_inf_sdiff
-- TODO: in distributive lattices, relative complements are unique when they exist
theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm]
| rw [inf_comm] at i | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm]
| Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
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