An Extension of Stolarsky Means to the Multivariable Case

Abstract

We give an extension of well-known Stolarsky means to the multivariable case in a simple and applicable way. Some basic inequalities concerning this matter are also established with applications in Analysis and Probability Theory.

1. Introduction

There is a huge amount of papers investigating properties of the so-called Stolarsky (or extended) two-parametric mean value, defined for positive values of 𝑥,𝑦, as

𝐸_𝑟,𝑠(𝑥,𝑦)∶=(𝑟(𝑥^𝑠−𝑦^𝑠)𝑠(𝑥^𝑟−𝑦^𝑟))^{1/(𝑠−𝑟)},𝑟𝑠(𝑟−𝑠)(𝑥−𝑦)≠0. (1.1)

𝐸 means can be continuously extended on the domain

{(𝑟,𝑠;𝑥,𝑦)∣𝑟,𝑠∈ℝ;𝑥,𝑦∈ℝ₊} (1.2) by the following:

𝐸_𝑟,𝑠((𝑥,𝑦)=𝑟(𝑥^𝑠−𝑦^𝑠)s(𝑥^𝑟−𝑦^𝑟))^{1/(𝑠−𝑟)}1𝑟𝑠(𝑟−𝑠)≠0;exp(−𝑠+𝑥^𝑠log𝑥−𝑦^𝑠log𝑦𝑥^𝑠−𝑦^𝑠(𝑥),𝑟=𝑠≠0;_𝑠−𝑦^𝑠)𝑠(log𝑥−log𝑦)^1/𝑠√,𝑠≠0,𝑟=0;𝑥𝑦,𝑟=𝑠=0;𝑥,𝑦=𝑥>0, (1.3) and in this form are introduced by Keneth Stolarsky in [1].

Most of the classical two-variable means are special cases of the class 𝐸. For example, 𝐸_1,2=(𝑥+𝑦)/2 is the arithmetic mean, 𝐸_0,0=√𝑥𝑦 is the geometric mean, 𝐸_0,1=(𝑥−𝑦)/(log𝑥−log𝑦) is the logarithmic mean, 𝐸_1,1=(𝑥^𝑥/𝑦^𝑦)^{1/(𝑥−𝑦)}/𝑒 is the identric mean, and so forth. More generally, the 𝑟th power mean ((𝑥^𝑟+𝑦^𝑟)/2)^1/𝑟 is equal to 𝐸_𝑟,2𝑟.

Recently, several papers are produced trying to define an extension of the class 𝐸 to 𝑛,𝑛>2 variables. Unfortunately, this is done in a highly artificial mode (cf. [2–4]), without a practical background. Here is an illustration of this point; recently Merikowski [4] has proposed the following generalization of the Stolarsky mean 𝐸_𝑟,𝑠 to several variables:

𝐸_𝑟,𝑠(𝑋)∶=[𝐿(𝑋^𝑠)𝐿(𝑋^𝑟)]^{1/(𝑠−𝑟)},𝑟≠𝑠, (1.4) where 𝑋=(𝑥₁,…,𝑥_𝑛) is an 𝑛-tuple of positive numbers and

𝐿(𝑋^𝑠)∶=(𝑛−1)!_𝐸𝑛−1^𝑛_𝑖=1𝑥^𝑠𝑢𝑖_𝑖𝑑𝑢₁⋯𝑑𝑢_𝑛−1. (1.5)

The symbol 𝐸_𝑛−1 stands for the Euclidean simplex which is defined by

𝐸_𝑛−1∶={(𝑢₁,…,𝑢_𝑛−1)∶𝑢_𝑖≥0,1≤𝑖≤𝑛−1;𝑢₁+⋯+𝑢_𝑛−1≤1}. (1.6)

In this paper, we give another attempt to generalize Stolarsky means to the multivariable case in a simple and applicable way. The proposed task can be accomplished by founding a “weighted” variant of the class 𝐸, wherefrom the mentioned generalization follows naturally.

In the sequel, we will need notions of the weighted geometric mean 𝐺=𝐺(𝑝,𝑞;𝑥,𝑦) and weighted 𝑟th power mean 𝑆_𝑟=𝑆_𝑟(𝑝,𝑞;𝑥,𝑦), defined by

𝐺∶=𝑥^𝑝𝑦^𝑞;𝑆_𝑟∶=(𝑝𝑥^𝑟+𝑞𝑦^𝑟)^1/𝑟, (1.7) where

𝑝,𝑞,𝑥,𝑦∈ℝ₊;𝑝+𝑞=1;𝑟∈ℝ/{0}. (1.8) Note that (𝑆_𝑟)^𝑟>(𝐺)^𝑟 for 𝑥≠𝑦,𝑟≠0, and lim_𝑟→0𝑆_𝑟=𝐺.

1.1. Weighted Stolarsky Means

We introduce here a class 𝑊 of weighted two-parameters means which includes the Stolarsky class 𝐸 as a particular case. Namely, for 𝑝,𝑞,𝑥,𝑦∈𝑅₊,𝑝+𝑞=1,𝑟𝑠(𝑟−𝑠)(𝑥−𝑦)≠0, we define

𝑊=𝑊_𝑟,𝑠𝑟(𝑝,𝑞;𝑥,𝑦)∶=(²𝑠²(𝑆_𝑠)^𝑠−(𝐺)^𝑠(𝑆_𝑟)^𝑟−(𝐺)^𝑟)^{1/(𝑠−𝑟)}𝑟=(²𝑠²𝑝𝑥^𝑠+𝑞𝑦^𝑠−𝑥^𝑝𝑠𝑦^𝑞𝑠𝑝𝑥^𝑟+𝑞𝑦^𝑟−𝑥^𝑝𝑟𝑦^𝑞𝑟)^{1/(𝑠−𝑟)}. (1.9) Various properties concerning the means 𝑊 can be established; some of them are the following:

𝑊_𝑟,𝑠(𝑝,𝑞;𝑥,𝑦)=𝑊_𝑠,𝑟𝑊(𝑝,𝑞;𝑥,𝑦);_𝑟,𝑠(𝑝,𝑞;𝑥,𝑦)=𝑊_𝑟,𝑠(𝑞,𝑝;𝑦,𝑥);𝑊_𝑟,𝑠(𝑝,𝑞;𝑦,𝑥)=𝑥𝑦𝑊_𝑟,𝑠(𝑝,𝑞;𝑥⁻¹,𝑦⁻¹𝑊);_{𝑎𝑟,𝑎𝑠}(𝑝,𝑞;𝑥,𝑦)=(𝑊_𝑟,𝑠(𝑝,𝑞;𝑥^𝑎,𝑦^𝑎))^1/𝑎,𝑎≠0. (1.10) Note that

𝑊_2𝑟,2𝑠(12,12𝑟;𝑥,𝑦)=(²𝑠²𝑥^2𝑠+𝑦^2𝑠−2(√𝑥𝑦)^2𝑠𝑥^2𝑟+𝑦^2𝑟−2(√𝑥𝑦)^2𝑟)^{1/2(𝑠−𝑟)}𝑟=(²𝑠²(𝑥^𝑠−𝑦^𝑠)²(𝑥^𝑟−𝑦^𝑟)²)^{1/2(𝑠−𝑟)}=𝐸(𝑟,𝑠;𝑥,𝑦). (1.11)

In the same manner, we get

𝑊_𝑟,𝑠(23,13;𝑥³,𝑦³)=(2𝑥^𝑠+𝑦^𝑠2𝑥^𝑟+𝑦^𝑟)^{1/(𝑠−𝑟)}(𝐸(𝑟,𝑠;𝑥,𝑦))²;𝑊_𝑟,𝑠(34,14;𝑥⁴,𝑦⁴)=(3𝑥^2𝑠−(𝑥𝑦)^𝑠+𝑦^2𝑠3𝑥^2𝑟−(𝑥𝑦)^𝑟+𝑦^2𝑟)^{1/(𝑠−𝑟)}(𝐸(𝑟,𝑠;𝑥,𝑦))². (1.12)

The weighted means from the class 𝑊 can be extended continuously to the domain

𝐷={(𝑟,𝑠;𝑥,𝑦)∣𝑟,𝑠∈ℝ;𝑥,𝑦∈ℝ₊}. (1.13) This extension is given by

𝑊_𝑟,𝑠=(𝑟(𝑝,𝑞;𝑥,𝑦)₂𝑠²𝑝𝑥^𝑠+𝑞𝑦^𝑠−𝑥^𝑝𝑠𝑦^𝑞𝑠𝑝𝑥^𝑟+𝑞𝑦^𝑟−𝑥^𝑝𝑟𝑦^𝑞𝑟)^{1/(𝑠−𝑟)},𝑟𝑠(𝑟−𝑠)(𝑥−𝑦)≠0;(2𝑝𝑥^𝑠+𝑞𝑦^𝑠−𝑥^𝑝𝑠𝑦^𝑞𝑠𝑝𝑞𝑠²log²)(𝑥/𝑦)^1/𝑠,𝑠(𝑥−𝑦)≠0,𝑟=0;exp(−2𝑠+𝑝𝑥^𝑠log𝑥+𝑞𝑦^𝑠log𝑦−(𝑝log𝑥+𝑞log𝑦)𝑥^𝑝𝑠𝑦^𝑞𝑠𝑝𝑥^𝑠+𝑞𝑦^𝑠−𝑥^𝑝𝑠𝑦^𝑞𝑠𝑥),𝑠(𝑥−𝑦)≠0,𝑟=𝑠;_(𝑝+1)/3𝑦^(𝑞+1)/3,𝑥≠𝑦,𝑟=𝑠=0;𝑥,𝑥=𝑦. (1.14)

Note that those means are homogeneous of order 1, that is, 𝑊_𝑟,𝑠(𝑝,𝑞;𝑡𝑥,𝑡𝑦)=𝑡𝑊_𝑟,𝑠(𝑝,𝑞;𝑥,𝑦),𝑡>0, symmetric in 𝑟,𝑠, 𝑊_𝑟,𝑠(𝑝,𝑞;𝑥,𝑦)=𝑊_𝑠,𝑟(𝑝,𝑞;𝑥,𝑦) but are not symmetric in 𝑥,𝑦 unless 𝑝=𝑞=1/2.

1.2. Multivariable Case

A natural generalization of weighted Stolarsky means to the multivariable case gives

𝑊_𝑟,𝑠(𝑟(𝐩;𝐱)=²(∑𝑝_𝑖𝑥^𝑠_𝑖−(∏𝑥^𝑝𝑖_𝑖)^𝑠)𝑠²(∑𝑝_𝑖𝑥^𝑟_𝑖−(∏𝑥^𝑝𝑖_𝑖)^𝑟))^{1/(𝑠−𝑟)}(2,𝑟𝑠(𝑠−𝑟)≠0;𝑠²∑𝑝_𝑖𝑥^𝑠_𝑖−(∏𝑥^𝑝𝑖_𝑖)^𝑠∑𝑝_𝑖log²𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)²)^1/𝑠,𝑟=0,𝑠≠0;exp(−2𝑠+∑𝑝_𝑖𝑥^𝑠_𝑖log𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)(∏𝑥^𝑝𝑖_𝑖)^𝑠∑𝑝_𝑖𝑥^𝑠_𝑖−(∏𝑥^𝑝𝑖_𝑖)^𝑠),𝑟=𝑠≠0;exp(∑𝑝_𝑖log³𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)³3(∑𝑝_𝑖log²𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)²)),𝑟=𝑠=0, (1.15) where 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛₊,𝑛≥2, 𝐩 is an arbitrary positive weight sequence associated with 𝐱 and 𝑊_𝑟,𝑠(𝐩;𝐱₀)=𝑎 for 𝐱₀=(𝑎,𝑎,…,𝑎).

We also write ∑(⋅),∏(⋅) instead of ∑^𝑛₁(⋅),∏^𝑛₁(⋅).

The above formulae are obtained by an appropriate limit process, implying continuity.

For example, applying

𝑡^𝑠𝑠=1+𝑠log𝑡+²2log²𝑠𝑡+³6log³𝑡+𝑜(𝑠³)(𝑠→0), (1.16) we get

𝑊_0,0(𝐩;𝐱)=lim_𝑠→0𝑊_𝑠,0(𝐩;𝐱)=lim_𝑠→0(2𝑠²∑𝑝_𝑖𝑥^𝑠_𝑖−(∏𝑥^𝑝𝑖_𝑖)^𝑠∑𝑝_𝑖log²𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)²)^1/𝑠=lim_𝑠→0(2𝑠²(∑𝑝_𝑖log²𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)²)×((𝑝_𝑖+𝑠𝑝_𝑖log𝑥_𝑖𝑠+(²2)𝑝_𝑖log²𝑥_𝑖𝑠+(³6)𝑝_𝑖log³𝑥_𝑖)−(𝑝_𝑖+𝑠log(𝑥^𝑝𝑖_𝑖𝑠)+(²2)log²(𝑥^𝑝𝑖_𝑖)𝑠+(³6)log³(𝑥^𝑝𝑖_𝑖))+𝑜(𝑠³)))^1/𝑠=lim_𝑠→0(1+∑𝑝_𝑖log³𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)³3(∑𝑝_𝑖log²𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)²)𝑠(1+𝑜(1)))^1/𝑠=exp(∑𝑝_𝑖log³𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)³3(∑𝑝_𝑖log²𝑥_𝑖−(∑𝑝_𝑖log𝑥_𝑖)²)). (1.17)

Remark 1.1. Analogously to the former considerations, one can define a class of Stolarsky means in 𝑛 variables 𝐸_𝑟,𝑠(𝐱;𝑛) as 𝐸_𝑟,𝑠(𝐱;𝑛)∶=𝑊_{𝑛𝑟,𝑛𝑠}(𝐩₀,𝐱), (1.18) where 𝐩₀={1/𝑛}^𝑛₁.
Therefore, 𝐸_𝑟,𝑠𝑟(𝐱;𝑛)=(²𝑠²∑^𝑛₁𝑥_𝑖^𝑛𝑠−𝑛∏^𝑛₁𝑥^𝑠_𝑖∑^𝑛₁𝑥_𝑖^𝑛𝑟−𝑛∏^𝑛₁𝑥^𝑟_𝑖)^{1/𝑛(𝑠−𝑟)},𝑟𝑠(𝑟−𝑠)≠0. (1.19) Details are left to the readers.

2. Results

The following basic assertion is of importance.

Proposition 2.1. The expressions 𝑊_𝑟,𝑠(𝐩;𝐱) are actual means, that is, for arbitrary weight sequence 𝐩 one has min{𝑥₁,𝑥₂…,𝑥_𝑛}≤𝑊_𝑟,𝑠(𝐩;𝐱)≤max{𝑥₁,𝑥₂,…,𝑥_𝑛}. (2.1)

Our main result is contained in the following.

Proposition 2.2. The means 𝑊_𝑟,𝑠(𝐩,𝐱) are monotone increasing in both variables 𝑟 and 𝑠.

Passing to the continuous variable case, we get the following definition of the class 𝑊_𝑟,𝑠(𝑝,𝑥).

Assuming that all integrals exist,

𝑊_𝑟,𝑠=(𝑟(𝐩,𝐱)₂(∫𝑝(𝑡)𝑥^𝑠(𝑡)𝑑𝑡−exp(𝑠∫𝑝(𝑡)log𝑥(𝑡)𝑑𝑡))𝑠²(∫𝑝(𝑡)𝑥^𝑟)(𝑡)𝑑𝑡−exp(𝑟∫𝑝(𝑡)log𝑥(𝑡)𝑑𝑡))^{1/(𝑠−𝑟)}(2,𝑟𝑠(𝑠−𝑟)≠0;𝑠²∫𝑝(𝑡)𝑥^𝑠(𝑡)𝑑𝑡−exp(𝑠∫𝑝(𝑡)log𝑥(𝑡)𝑑𝑡)∫𝑝(𝑡)log²𝑥(𝑡)𝑑𝑡−(∫𝑝(𝑡)log𝑥(𝑡)𝑑𝑡)²)^1/𝑠,𝑟=0,𝑠≠0;exp(−2𝑠+∫𝑝(𝑡)𝑥^𝑠(𝑡)log𝑥(𝑡)𝑑𝑡−(∫𝑝(𝑡)log𝑥(𝑡)𝑑𝑡)exp(𝑠∫𝑝(𝑡)log𝑥(𝑡)𝑑𝑡)∫𝑝(𝑡)𝑥^𝑠∫(𝑡)𝑑𝑡−exp(𝑠∫𝑝(𝑡)log𝑥(𝑡)𝑑𝑡)),𝑟=𝑠≠0;exp(𝑝(𝑡)log³𝑥(𝑡)𝑑𝑡−(∫𝑝(𝑡)log𝑥(𝑡)𝑑𝑡)³3(∫𝑝(𝑡)log²𝑥(𝑡)𝑑𝑡−(∫𝑝(𝑡)log𝑥(𝑡)𝑑𝑡)²)),𝑟=𝑠=0, (2.2) where 𝑥(𝑡) is a positive integrable function and 𝑝(𝑡) is a nonnegative function with ∫𝑝(𝑡)𝑑𝑡=1.

From our former considerations, a very applicable assertion follows.

Proposition 2.3. 𝑊_𝑟,𝑠(𝐩,𝐱) is monotone increasing in either 𝑟 or 𝑠.

3. Applications

3.1. Applications in Analysis

As an illustration of the above, we give the following proposition.

Proposition 3.1. The function 𝑤(𝑠), defined by (𝑤(𝑠)∶=12(𝜋𝑠)²(Γ(1+𝑠)−𝑒^−𝛾𝑠))^1/𝑠,𝑠≠0;exp(−𝛾−4𝜉(3)𝜋²),𝑠=0, (3.1) is monotone increasing for 𝑠∈(−1,∞).
In particular, for 𝑠∈(−1,1), one has Γ(1−𝑠)𝑒^−𝛾𝑠+Γ(1+𝑠)𝑒^𝛾𝑠−𝜋𝑠sin(𝜋𝑠)≤1−(𝜋𝑠)⁴,144 (3.2) where Γ(⋅),𝜉(⋅),𝛾 stands for the Gamma function, Zeta function, and Euler's constant, respectively.

3.2. Applications in Probability Theory

For a random variable 𝑋 and an arbitrary probability distribution with support on (−∞,+∞), it is well known that

𝐸𝑒^𝑋≥𝑒^𝐸𝑋. (3.3)

Denoting the central moment of order 𝑘 by 𝜇_𝑘=𝜇_𝑘(𝑋)∶=𝐸(𝑋−𝐸𝑋)^𝑘, we improve this inequality to the following propsositions.

Proposition 3.2. For an arbitrary probability law with support on ℝ, one has 𝐸𝑒^𝑋𝜇≥(1+(²2𝜇)exp(³3𝜇₂))𝑒^𝐸𝑋. (3.4)

Proposition 3.3. One also has that (𝐸𝑒^𝑠𝑋−𝑒^𝑠𝐸𝑋𝑠²𝜎²_𝑋)/2^1/𝑠 (3.5) is monotone increasing in 𝑠.

3.3. Shifted Stolarsky Means

Especially interesting is studying the shifted Stolarsky means 𝐸^∗, defined by

𝐸^∗_𝑟,𝑠(𝑥,𝑦)∶=lim_𝑝→0+𝑊_𝑟,𝑠(𝑝,𝑞;𝑥,𝑦). (3.6) Their analytic continuation to the whole (𝑟,𝑠) plane is given by

𝐸^∗_𝑟,𝑠(𝑟(𝑥,𝑦)=²(𝑥^𝑠−𝑦^𝑠(1+𝑠log(𝑥/𝑦)))𝑠²(𝑥^𝑟−𝑦^𝑟)(1+𝑟log(𝑥/𝑦)))^{1/(𝑠−𝑟)}(2,𝑟𝑠(𝑟−𝑠)(𝑥−𝑦)≠0;𝑠²𝑥^𝑠−𝑦^𝑠(1+𝑠log(𝑥/𝑦))log²)(𝑥/𝑦)^1/𝑠,𝑠(𝑥−𝑦)≠0,𝑟=0;exp(−2𝑠+(𝑥^𝑠−𝑦^𝑠)log𝑥−𝑠𝑦^𝑠log𝑦log(𝑥/𝑦)𝑥^𝑠−𝑦^𝑠𝑥(1+𝑠log(𝑥/𝑦))),𝑠(𝑥−𝑦)≠0,𝑟=𝑠;_1/3𝑦^2/3,𝑟=𝑠=0;𝑥,𝑥=𝑦. (3.7)

Main results concerning the means 𝐸^∗ are contained in the following propositions.

Proposition 3.4. Means 𝐸^∗_𝑟,𝑠(𝑥,𝑦) are monotone increasing in either 𝑟 or 𝑠 for each fixed 𝑥,𝑦∈ℝ⁺.

Proposition 3.5. Means 𝐸^∗_𝑟,𝑠(𝑥,𝑦) are monotone increasing in either 𝑥 or y for each 𝑟,𝑠∈ℝ.

A well known result of Qi ([5]) states that the means 𝐸_𝑟,𝑠(𝑥,𝑦) are logarithmically concave for each fixed 𝑥,𝑦>0 and 𝑟,𝑠∈[0,+∞); also, they are logarithmically convex for 𝑟,𝑠∈(−∞,0].

According to this, we propose the following proposition.

Open Question
Is there any compact interval 𝐼,𝐼⊂ℝ such that the means 𝐸^∗_𝑟,𝑠(𝑥,𝑦) are logarithmically convex (concave) for 𝑟,𝑠∈𝐼 and each 𝑥,𝑦∈ℝ⁺?

A partial answer to this problem is given in what follows.

Proposition 3.6. On any interval I which includes zero and 𝑟,𝑠∈𝐼,
(𝑖) 𝐸^∗_𝑟,𝑠(𝑥,𝑦) are not logarithmically convex (concave);
(𝑖𝑖) 𝑊_𝑟,𝑠(𝑝,𝑞;𝑥,𝑦) are logarithmically convex (concave) if and only if 𝑝=𝑞=1/2.

4. Proofs

For the proof of Proposition 2.1, we apply the following assertion on Jensen functionals 𝐽_𝑓(𝐩,𝐱) from [6].

Theorem 4.1. Let 𝑓,𝑔∶𝐼→ℝ be twice continuously differentiable functions. Assume that 𝑔 is strictly convex and 𝜙 is a continuous and strictly monotonic function on 𝐼. Then the expression 𝜙⁻¹(𝐽_𝑛(𝐩,𝐱;𝑓)𝐽_𝑛(𝐩,𝐱;𝑔))(𝑛≥2) (4.1) represents a mean value of the numbers 𝑥₁,…,𝑥_𝑛, that is, min{𝑥₁,…,𝑥_𝑛}≤𝜙⁻¹(𝐽_𝑛(𝐩,𝐱;𝑓)𝐽_𝑛(𝐩,𝐱;𝑔))≤max{𝑥₁,…,𝑥_𝑛} (4.2) if and only if the relation 𝑓^′′(𝑡)=𝜙(𝑡)𝑔^′′(𝑡) (4.3) holds for each 𝑡∈𝐼.

Recall that the Jensen functional 𝐽_𝑛(𝐩,𝐱;𝑓) is defined on an interval 𝐼,𝐼⊆ℝ by

𝐽_𝑛(𝐩,𝐱;𝑓)∶=^𝑛₁𝑝_𝑖𝑓(𝑥_𝑖)−𝑓(^𝑛₁𝑝_𝑖𝑥_𝑖), (4.4) where 𝑓∶𝐼→ℝ, 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝐼^𝑛, and 𝐩={𝑝_𝑖}^𝑛₁ is a positive weight sequence.

The famous Jensen's inequality asserts that

𝐽_𝑛(𝐩,𝐱;𝑓)≥0, (4.5) whenever 𝑓 is a (strictly) convex function on 𝐼, with the equality case if and only if 𝑥₁=𝑥₂=⋯=𝑥_𝑛.

Proof of Proposition 2.1. Define the auxiliary function ℎ_𝑠(𝑥) by ℎ_𝑠𝑒(𝑥)∶=^𝑠𝑥−𝑠𝑥−1𝑠²𝑥,𝑠≠0;₂2,𝑠=0. (4.6) Since ℎ^′_𝑠𝑒(𝑥)=^𝑠𝑥−1𝑠ℎ,𝑠≠0;𝑥,𝑠=0,_𝑠′′(𝑥)=𝑒^𝑠𝑥,𝑠∈ℝ, (4.7) we conclude that ℎ_𝑠(𝑥) is a continuously twice differentiable convex function on ℝ.
Denoting 𝑓(𝑡)∶=ℎ_𝑠(𝑡),𝑔(𝑡)∶=ℎ_𝑟(𝑡), we realize that the condition (4.3) of Theorem 4.1 is fulfilled with 𝜙(𝑡)=𝑒^{(𝑠−𝑟)𝑡}. Hence, applying Theorem 4.1, we obtain that log𝑊_𝑟,𝑠(𝐩,𝑒^𝐱) represents a mean value, which is equivalent to the assertion of Proposition 2.1.

Proof of Proposition 2.2. We prove first a global theorem concerning log-convexity of the Jensen's functional with a parameter, which can be very usable (cf. [7]).Theorem 4.2. Let 𝑓_𝑠(𝑥) be a twice continuously differentiable function in 𝑥 with a parameter 𝑠. If 𝑓_𝑠^′′(𝑥) is log-convex in 𝑠 for 𝑠∈𝐼∶=(𝑎,𝑏);𝑥∈𝐾∶=(𝑐,𝑑), then the Jensen functional 𝐽_𝑓(𝑤,𝑥;𝑠)=𝐽(𝑠)∶=𝑤_𝑖𝑓_𝑠(𝑥_𝑖)−𝑓_𝑠(𝑤_𝑖𝑥_𝑖), (4.8) is log-convex in 𝑠 for 𝑠∈𝐼,𝑥_𝑖∈𝐾,𝑖=1,2,…, where 𝑤={𝑤_𝑖} is any positive weight sequence.At the beginning, we need some preliminary lemmas. Lemma 4.3. A positive function 𝑓 is log-convex on 𝐼 if and only if the relation 𝑓(𝑠)𝑢²+2𝑓(𝑠+𝑡2)𝑢𝑤+𝑓(𝑡)𝑤²≥0 (4.9) holds for each real 𝑢,𝑤 and 𝑠,𝑡∈𝐼. This assertion is nothing more than the discriminant test for the nonnegativity of second-order polynomials. Other well known assertions are the following (cf [8, pages 74, 97-98]) lemmas.Lemma 4.4 (Jensen's inequality). If 𝑔(𝑥) is twice continuously differentiable and 𝑔^′′(𝑥)≥0 on 𝐾, then 𝑔(𝑥) is convex on 𝐾 and the inequality 𝑤_𝑖𝑔(𝑥_𝑖)−𝑔(𝑤_𝑖𝑥_𝑖)≥0 (4.10) holds for each 𝑥_𝑖∈𝐾,𝑖=1,2,…, and any positive weight sequence {𝑤_𝑖},∑𝑤_𝑖=1. Lemma 4.5. For a convex 𝑓, the expression 𝑓(𝑠)−𝑓(𝑟)𝑠−𝑟 (4.11) is increasing in both variables.Proof of Theorem 4.2. Consider the function 𝐹(𝑥) defined as 𝐹(𝑥)=𝐹(𝑢,𝑣,𝑠,𝑡;𝑥)∶=𝑢²𝑓_𝑠(𝑥)+2𝑢𝑣𝑓_{(𝑠+𝑡)/2}(𝑥)+𝑣²𝑓_𝑡(𝑥), (4.12) where 𝑢,𝑣∈ℝ;𝑠,𝑡∈𝐼 are real parameters independent of the variable 𝑥∈𝐾.
Since 𝐹^′′(𝑥)=𝑢²𝑓_𝑠^′′(𝑥)+2𝑢𝑣𝑓^′′_{(𝑠+𝑡)/2}(𝑥)+𝑣²𝑓_𝑡^′′(𝑥), (4.13) and by assuming 𝑓_𝑠^′′(𝑥) is log-convex in 𝑠, it follows from Lemma 4.3 that 𝐹^′′(𝑥)≥0,𝑥∈𝐾.
Therefore, by Lemma 4.4, we get 𝑤_𝑖𝐹(𝑥_𝑖)−𝐹(𝑤_𝑖𝑥_𝑖)≥0,𝑥_𝑖∈𝐾, (4.14) which is equivalent to 𝑢²𝐽(𝑠)+2𝑢𝑣𝐽(𝑠+𝑡2)+𝑣²𝐽(𝑡)≥0. (4.15)
According to Lemma 4.3 again, this is possible only if 𝐽(𝑠) is log-convex and the proof is done.
Now, the proof of Proposition 2.2 easily follows.
From the above, we see that ℎ_𝑠(𝑥) is twice continuously differentiable and that ℎ_𝑠^′′(𝑥) is a log-convex function for each real 𝑠,𝑥.
Applying Theorem 4.2, we conclude that the form Φ_ℎ(𝑤,𝑥;𝑠)=Φ(𝑠)∶=∑𝑤_𝑖𝑒^𝑠𝑥𝑖−𝑒^{𝑠∑𝑤𝑖𝑥𝑖}𝑠²,𝑠≠0,∑𝑤_𝑖𝑥²_𝑖−(∑𝑤_𝑖𝑥_𝑖)²2,𝑠=0, (4.16) is log-convex in 𝑠.
By Lemma 4.5, with 𝑓(𝑠)=logΦ(𝑠), we find out that logΦ(𝑠)−logΦ(𝑟)𝑠−𝑟=log(Φ(𝑠))Φ(𝑟)^{1/(𝑠−𝑟)} (4.17) is monotone increasing either in 𝑠 or 𝑟. Therefore, by changing variable 𝑥_𝑖→log𝑥_𝑖, we finally obtain the proof of Proposition 2.2.

Proof of Proposition 2.3. The assertion of Proposition 2.3 follows from Proposition 2.2 by the standard argument (cf. [8, pages 131–134]). Details are left to the reader.

Proof of Proposition 3.1. The proof follows putting 𝑥(𝑡)=𝑡,𝑝(𝑡)=𝑒^−𝑡,𝑡∈(0,+∞) and applying Proposition 2.2. with 𝑟=0. Corresponding integrals are ^∞₀𝑒^−𝑡log𝑡=−𝛾;^∞₀𝑒^−𝑡log²𝑡=𝛾²+𝜋²6;^∞₀𝑒^−𝑡log³𝑡=−𝛾³−𝛾𝜋²2−2𝜉(3), (4.18) with Γ(1−𝑠)Γ(1+𝑠)=𝜋𝑠.sin(𝜋𝑠) (4.19)

Proof of Proposition 3.2. By Proposition 2.3, we get 𝑊_0,1(𝐩,𝑒^𝐱)≥𝑊_0,0(𝐩,𝑒^𝐱), (4.20) that is, 𝐸𝑒^𝑋−𝑒^𝐸𝑋𝜇₂/2≥exp(𝐸𝑋³−(𝐸𝑋)³3𝜇₂). (4.21) Using the identity 𝐸𝑋³−(𝐸𝑋)³=𝜇₃+3𝜇₂𝐸𝑋, we obtain the proof of Proposition 3.2.

Proof of Proposition 3.3. This assertion is straightforward consequence of the fact that 𝑊_0,𝑠(𝐩,𝑒^𝐱) is monotone increasing in 𝑠.

Proof of Proposition 3.4. Direct consequence of Proposition 2.2.

Proof of Proposition 3.5. This is left as an easy exercise to the readers.

Proof of Proposition 3.6. We prove only part (ii). The proof of (i) goes along the same lines.
Suppose that 0∈(𝑎,𝑏)∶=𝐼 and that 𝐸_𝑟,𝑠(𝑝,𝑞;𝑥,𝑦) are log-convex (concave) for 𝑟,𝑠∈𝐼 and any fixed 𝑥,𝑦∈ℝ⁺. Then there should be an 𝑠,𝑠>0 such that 𝐹_𝑠(𝑝,𝑞;𝑥,𝑦)∶=𝑊_0,𝑠(𝑝,𝑞;𝑥,𝑦)𝑊_0,−𝑠(𝑝,𝑞;𝑥,𝑦)−(𝑊_0,0(𝑝,𝑞;𝑥,𝑦))² (4.22) is of constant sign for each 𝑥,𝑦>0.
Substituting (𝑥/𝑦)^𝑠∶=𝑒^𝑤,𝑤∈ℝ, after some calculations, we get that the above is equivalent to the assertion that 𝐹(𝑝,𝑞;𝑤) is of constant sign, where 𝐹(𝑝,𝑞;𝑤)∶=𝑝𝑒^𝑤+𝑞−𝑒^𝑝𝑤−𝑒^{(2/3)(1+𝑝)𝑤}(𝑝𝑒^−𝑤+𝑞−𝑒^−𝑝𝑤). (4.23) Developing in power series in 𝑤, we get 1𝐹(𝑝,𝑞;𝑤)=1620𝑝𝑞(1+𝑝)(2−𝑝)(1−2𝑝)𝑤⁵+𝑂(𝑤⁶). (4.24)
Therefore, 𝐹(𝑝,𝑞;𝑤) can be of constant sign for each 𝑤∈ℝ only if 𝑝=1/2(=𝑞).
Suppose now that 𝐼 is of the form 𝐼∶=[0,𝑎) or 𝐼∶=(−𝑎,0],𝑎>0. Then there should be an 𝑠,𝑠≠0,𝑠∈𝐼 such that 𝑊_0,0(𝑝,𝑞;𝑥,𝑦)𝑊_0,2𝑠(𝑝,𝑞;𝑥,𝑦)−(𝑊_0,𝑠(𝑝,𝑞;𝑥,𝑦))² (4.25) is of constant sign for each 𝑥,𝑦∈ℝ⁺.
Proceeding as before, this is equivalent to the assertion that 𝐺(𝑝,𝑞;𝑤) is of constant sign with 𝐺(𝑝,𝑞;𝑤)∶=𝑝³𝑞³𝑤⁶𝑒^{(2/3)(𝑝+1)𝑤}(𝑝𝑒^2𝑤+𝑞−𝑒^2𝑝𝑤)−(𝑝𝑒^𝑤+𝑞−𝑒^𝑝𝑤)⁴. (4.26) However, 2𝐺(𝑝,𝑞;𝑤)=𝑝405⁴𝑞⁴(1+𝑝)(1+𝑞)(𝑞−𝑝)𝑤¹¹+𝑂(𝑤¹²). (4.27)
Hence, we conclude that 𝐺(𝑝,𝑞;𝑤) can be of constant sign for sufficiently small 𝑤,𝑤∈ℝ only if 𝑝=𝑞=1/2. Combining this with Feng Qi theorem, the assertion from Proposition 3.6 follows.