mathlib documentation

analysis.mean_inequalities

Mean value inequalities

In this file we prove several inequalities, including AM-GM inequality, Young's inequality, Hölder inequality, and Minkowski inequality.

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.

Generalized mean inequality

The inequality says that for two non-negative vectors $w$ and $z$ with $\sum_{i\in s} w_i=1$ and $p ≤ q$ we have $$ \sqrt[p]{\sum_{i\in s} w_i z_i^p} ≤ \sqrt[q]{\sum_{i\in s} w_i z_i^q}. $$

Currently we only prove this inequality for $p=1$. As in the rest of mathlib, we provide different theorems for natural exponents (pow_arith_mean_le_arith_mean_pow), integer exponents (fpow_arith_mean_le_arith_mean_fpow), and real exponents (rpow_arith_mean_le_arith_mean_rpow and arith_mean_le_rpow_mean). In the first two cases we prove $$ \left(\sum_{i\in s} w_i z_i\right)^n ≤ \sum_{i\in s} w_i z_i^n $$ in order to avoid using real exponents. For real exponents we prove both this and standard versions.

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 can be used to prove Hölder's inequality (see below) but we use a different proof.

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 real, nnreal and ennreal.

There are at least two short proofs of this inequality. In one proof we prenormalize both vectors, then apply Young's inequality to each $a_ib_i$. We use a different proof deducing 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, nnreal and ennreal.

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

theorem real.geom_mean_le_arith_mean_weighted {ι : Type u} (s : finset ι) (w z : ι → ) :
(∀ (i : ι), i s0 w i)∑ (i : ι) in s, w i = 1(∀ (i : ι), i s0 z i)∏ (i : ι) in s, z i ^ w i ∑ (i : ι) in s, (w i) * z i

AM-GM inequality: the geometric mean is less than or equal to the arithmetic mean, weighted version for real-valued nonnegative functions.

theorem real.pow_arith_mean_le_arith_mean_pow {ι : Type u} (s : finset ι) (w z : ι → ) (hw : ∀ (i : ι), i s0 w i) (hw' : ∑ (i : ι) in s, w i = 1) (hz : ∀ (i : ι), i s0 z i) (n : ) :
(∑ (i : ι) in s, (w i) * z i) ^ n ∑ (i : ι) in s, (w i) * z i ^ n

theorem real.pow_arith_mean_le_arith_mean_pow_of_even {ι : Type u} (s : finset ι) (w z : ι → ) (hw : ∀ (i : ι), i s0 w i) (hw' : ∑ (i : ι) in s, w i = 1) {n : } :
even n(∑ (i : ι) in s, (w i) * z i) ^ n ∑ (i : ι) in s, (w i) * z i ^ n

theorem real.fpow_arith_mean_le_arith_mean_fpow {ι : Type u} (s : finset ι) (w z : ι → ) (hw : ∀ (i : ι), i s0 w i) (hw' : ∑ (i : ι) in s, w i = 1) (hz : ∀ (i : ι), i s0 < z i) (m : ) :
(∑ (i : ι) in s, (w i) * z i) ^ m ∑ (i : ι) in s, (w i) * z i ^ m

theorem real.rpow_arith_mean_le_arith_mean_rpow {ι : Type u} (s : finset ι) (w z : ι → ) (hw : ∀ (i : ι), i s0 w i) (hw' : ∑ (i : ι) in s, w i = 1) (hz : ∀ (i : ι), i s0 z i) {p : } :
1 p(∑ (i : ι) in s, (w i) * z i) ^ p ∑ (i : ι) in s, (w i) * z i ^ p

theorem real.arith_mean_le_rpow_mean {ι : Type u} (s : finset ι) (w z : ι → ) (hw : ∀ (i : ι), i s0 w i) (hw' : ∑ (i : ι) in s, w i = 1) (hz : ∀ (i : ι), i s0 z i) {p : } :
1 p∑ (i : ι) in s, (w i) * z i (∑ (i : ι) in s, (w i) * z i ^ p) ^ (1 / p)

theorem nnreal.geom_mean_le_arith_mean_weighted {ι : Type u} (s : finset ι) (w z : ι → ℝ≥0) :
∑ (i : ι) in s, w i = 1∏ (i : ι) in s, z i ^ (w i) ∑ (i : ι) in s, (w i) * z i

The geometric mean is less than or equal to the arithmetic mean, weighted version for nnreal-valued functions.

theorem nnreal.geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1(p₁ ^ w₁) * p₂ ^ w₂ w₁ * p₁ + w₂ * p₂

The geometric mean is less than or equal to the arithmetic mean, weighted version for two nnreal numbers.

theorem nnreal.geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1((p₁ ^ w₁) * p₂ ^ w₂) * p₃ ^ w₃ w₁ * p₁ + w₂ * p₂ + w₃ * p₃

theorem nnreal.geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1(((p₁ ^ w₁) * p₂ ^ w₂) * p₃ ^ w₃) * p₄ ^ w₄ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄

theorem nnreal.pow_arith_mean_le_arith_mean_pow {ι : Type u} (s : finset ι) (w z : ι → ℝ≥0) (hw' : ∑ (i : ι) in s, w i = 1) (n : ) :
(∑ (i : ι) in s, (w i) * z i) ^ n ∑ (i : ι) in s, (w i) * z i ^ n

Weighted generalized mean inequality, version sums over finite sets, with ℝ≥0-valued functions and natural exponent.

theorem nnreal.rpow_arith_mean_le_arith_mean_rpow {ι : Type u} (s : finset ι) (w z : ι → ℝ≥0) (hw' : ∑ (i : ι) in s, w i = 1) {p : } :
1 p(∑ (i : ι) in s, (w i) * z i) ^ p ∑ (i : ι) in s, (w i) * z i ^ p

Weighted generalized mean inequality, version for sums over finite sets, with ℝ≥0-valued functions and real exponents.

theorem nnreal.arith_mean_le_rpow_mean {ι : Type u} (s : finset ι) (w z : ι → ℝ≥0) (hw' : ∑ (i : ι) in s, w i = 1) {p : } :
1 p∑ (i : ι) in s, (w i) * z i (∑ (i : ι) in s, (w i) * z i ^ p) ^ (1 / p)

Weighted generalized mean inequality, version for sums over finite sets, with ℝ≥0-valued functions and real exponents.

theorem real.geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : } :
0 w₁0 w₂0 p₁0 p₂w₁ + w₂ = 1(p₁ ^ w₁) * p₂ ^ w₂ w₁ * p₁ + w₂ * p₂

theorem real.geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : } :
0 w₁0 w₂0 w₃0 p₁0 p₂0 p₃w₁ + w₂ + w₃ = 1((p₁ ^ w₁) * p₂ ^ w₂) * p₃ ^ w₃ w₁ * p₁ + w₂ * p₂ + w₃ * p₃

theorem real.geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : } :
0 w₁0 w₂0 w₃0 w₄0 p₁0 p₂0 p₃0 p₄w₁ + w₂ + w₃ + w₄ = 1(((p₁ ^ w₁) * p₂ ^ w₂) * p₃ ^ w₃) * p₄ ^ w₄ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄

theorem real.young_inequality_of_nonneg {a b p q : } :
0 a0 bp.is_conjugate_exponent qa * b a ^ p / p + b ^ q / q

Young's inequality, a version for nonnegative real numbers.

theorem real.young_inequality (a b : ) {p q : } :
p.is_conjugate_exponent qa * b abs a ^ p / p + abs b ^ q / q

Young's inequality, a version for arbitrary real numbers.

theorem nnreal.young_inequality (a b : ℝ≥0) {p q : ℝ≥0} :
1 < p1 / p + 1 / q = 1a * b a ^ p / p + b ^ q / q

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 nnreal.inner_le_Lp_mul_Lq {ι : Type u} (s : finset ι) (f g : ι → ℝ≥0) {p q : } :
p.is_conjugate_exponent q∑ (i : ι) in s, (f i) * g i ((∑ (i : ι) in s, f i ^ p) ^ (1 / p)) * (∑ (i : ι) 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 nnreal.is_greatest_Lp {ι : Type u} (s : finset ι) (f : ι → ℝ≥0) {p q : } :
p.is_conjugate_exponent qis_greatest ((λ (g : ι → ℝ≥0), ∑ (i : ι) in s, (f i) * g i) '' {g : ι → ℝ≥0 | ∑ (i : ι) in s, g i ^ q 1}) ((∑ (i : ι) in s, f i ^ p) ^ (1 / p))

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 nnreal.Lp_add_le {ι : Type u} (s : finset ι) (f g : ι → ℝ≥0) {p : } :
1 p(∑ (i : ι) in s, (f i + g i) ^ p) ^ (1 / p) (∑ (i : ι) in s, f i ^ p) ^ (1 / p) + (∑ (i : ι) 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 nnreal-valued functions.

theorem real.inner_le_Lp_mul_Lq {ι : Type u} (s : finset ι) (f g : ι → ) {p q : } :
p.is_conjugate_exponent q∑ (i : ι) in s, (f i) * g i ((∑ (i : ι) in s, abs (f i) ^ p) ^ (1 / p)) * (∑ (i : ι) in s, abs (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 real-valued functions.

theorem real.Lp_add_le {ι : Type u} (s : finset ι) (f g : ι → ) {p : } :
1 p(∑ (i : ι) in s, abs (f i + g i) ^ p) ^ (1 / p) (∑ (i : ι) in s, abs (f i) ^ p) ^ (1 / p) + (∑ (i : ι) in s, abs (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 real-valued functions.

theorem real.inner_le_Lp_mul_Lq_of_nonneg {ι : Type u} (s : finset ι) {f g : ι → } {p q : } :
p.is_conjugate_exponent q(∀ (i : ι), i s0 f i)(∀ (i : ι), i s0 g i)∑ (i : ι) in s, (f i) * g i ((∑ (i : ι) in s, f i ^ p) ^ (1 / p)) * (∑ (i : ι) 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 real-valued nonnegative functions.

theorem real.Lp_add_le_of_nonneg {ι : Type u} (s : finset ι) {f g : ι → } {p : } :
1 p(∀ (i : ι), i s0 f i)(∀ (i : ι), i s0 g i)(∑ (i : ι) in s, (f i + g i) ^ p) ^ (1 / p) (∑ (i : ι) in s, f i ^ p) ^ (1 / p) + (∑ (i : ι) 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 real-valued nonnegative functions.

theorem ennreal.inner_le_Lp_mul_Lq {ι : Type u} (s : finset ι) (f g : ι → ennreal) {p q : } :
p.is_conjugate_exponent q∑ (i : ι) in s, (f i) * g i ((∑ (i : ι) in s, f i ^ p) ^ (1 / p)) * (∑ (i : ι) 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 ennreal-valued functions.

theorem ennreal.Lp_add_le {ι : Type u} (s : finset ι) (f g : ι → ennreal) {p : } :
1 p(∑ (i : ι) in s, (f i + g i) ^ p) ^ (1 / p) (∑ (i : ι) in s, f i ^ p) ^ (1 / p) + (∑ (i : ι) 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 ennreal valued nonnegative functions.