# An Extension of Stolarsky Means to the Multivariable Case

Authors: Slavko Simic
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

---

## 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.


---

## Body

## 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 ofx,y, as(1.1)Er,s(x,y):=(r(xs-ys)s(xr-yr))1/(s-r),rs(r-s)(x-y)≠0.E means can be continuously extended on the domain(1.2){(r,s;x,y)∣r,s∈ℝ;x,y∈ℝ+}
by the following:(1.3)Er,s(x,y)={(r(xs-ys)s(xr-yr))1/(s-r)rs(r-s)≠0;exp(-1s+xslogx-yslogyxs-ys),r=s≠0;(xs-yss(logx-logy))1/s,s≠0,r=0;xy,r=s=0;x,y=x>0,
and in this form are introduced by Keneth Stolarsky in [1].Most of the classical two-variable means are special cases of the classE. For example, E1,2=(x+y)/2 is the arithmetic mean, E0,0=xy is the geometric mean, E0,1=(x-y)/(logx-logy) is the logarithmic mean, E1,1=(xx/yy)1/(x-y)/e is the identric mean, and so forth. More generally, the rth power mean ((xr+yr)/2)1/r is equal to Er,2r.Recently, several papers are produced trying to define an extension of the classE 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 Er,s to several variables:(1.4)Er,s(X):=[L(Xs)L(Xr)]1/(s-r),r≠s,
where X=(x1,…,xn) is an n-tuple of positive numbers and(1.5)L(Xs):=(n-1)!∫En-1∏i=1nxisuidu1⋯dun-1.The symbolEn-1 stands for the Euclidean simplex which is defined by(1.6)En-1:={(u1,…,un-1):ui≥0,1≤i≤n-1;u1+⋯+un-1≤1}.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 classE, wherefrom the mentioned generalization follows naturally.In the sequel, we will need notions of the weighted geometric meanG=G(p,q;x,y) and weighted rth power mean Sr=Sr(p,q;x,y), defined by(1.7)G:=xpyq;Sr:=(pxr+qyr)1/r,
where(1.8)p,q,x,y∈ℝ+;p+q=1;r∈ℝ/{0}.
Note that (Sr)r>(G)r for x≠y,r≠0, and limr→0Sr=G.


### 1.1. Weighted Stolarsky Means

We introduce here a classW 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(1.9)W=Wr,s(p,q;x,y):=(r2s2(Ss)s-(G)s(Sr)r-(G)r)1/(s-r)=(r2s2pxs+qys-xpsyqspxr+qyr-xpryqr)1/(s-r).
Various properties concerning the means W can be established; some of them are the following:(1.10)Wr,s(p,q;x,y)=Ws,r(p,q;x,y);Wr,s(p,q;x,y)=Wr,s(q,p;y,x);Wr,s(p,q;y,x)=xyWr,s(p,q;x-1,y-1);War,as(p,q;x,y)=(Wr,s(p,q;xa,ya))1/a,a≠0.
Note that(1.11)W2r,2s(12,12;x,y)=(r2s2x2s+y2s-2(xy)2sx2r+y2r-2(xy)2r)1/2(s-r)=(r2s2(xs-ys)2(xr-yr)2)1/2(s-r)=E(r,s;x,y).In the same manner, we get(1.12)Wr,s(23,13;x3,y3)=(2xs+ys2xr+yr)1/(s-r)(E(r,s;x,y))2;Wr,s(34,14;x4,y4)=(3x2s-(xy)s+y2s3x2r-(xy)r+y2r)1/(s-r)(E(r,s;x,y))2.The weighted means from the classW can be extended continuously to the domain(1.13)D={(r,s;x,y)∣r,s∈ℝ;x,y∈ℝ+}.
This extension is given by(1.14)Wr,s(p,q;x,y)={(r2s2pxs+qys-xpsyqspxr+qyr-xpryqr)1/(s-r),rs(r-s)(x-y)≠0;(2pxs+qys-xpsyqspqs2log2(x/y))1/s,s(x-y)≠0,r=0;exp(-2s+pxslogx+qyslogy-(plogx+qlogy)xpsyqspxs+qys-xpsyqs),s(x-y)≠0,r=s;x(p+1)/3y(q+1)/3,x≠y,r=s=0;x,x=y.Note that those means are homogeneous of order1, that is, Wr,s(p,q;tx,ty)=tWr,s(p,q;x,y),t>0, symmetric in r,s, Wr,s(p,q;x,y)=Ws,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(1.15)Wr,s(p;x)={(r2(∑pixis-(∏xipi)s)s2(∑pixir-(∏xipi)r))1/(s-r),rs(s-r)≠0;(2s2∑pixis-(∏xipi)s∑pilog2xi-(∑pilogxi)2)1/s,r=0,s≠0;exp(-2s+∑pixislogxi-(∑pilogxi)(∏xipi)s∑pixis-(∏xipi)s),r=s≠0;exp(∑pilog3xi-(∑pilogxi)33(∑pilog2xi-(∑pilogxi)2)),r=s=0,
where x=(x1,x2,…,xn)∈ℝ+n,n≥2, p is an arbitrary positive weight sequence associated with x and Wr,s(p;x0)=a for x0=(a,a,…,a).We also write∑(·),∏(·) instead of ∑1n(·),∏1n(·).The above formulae are obtained by an appropriate limit process, implying continuity.For example, applying(1.16)ts=1+slogt+s22log2t+s36log3t+o(s3)(s→0),
we get(1.17)W0,0(p;x)=lims→0Ws,0(p;x)=lims→0(2s2∑pixis-(∏xipi)s∑pilog2xi-(∑pilogxi)2)1/s=lims→0(+(s36)log3(∏xipi))+o(s3))2s2(∑pilog2xi-(∑pilogxi)2)×((∑pi+s∑pilogxi+(s22)∑pilog2xi+(s36)∑pilog3xi)-(∑pi+slog(∏xipi)+(s22)log2(∏xipi)+(s36)log3(∏xipi))+o(s3))(∑pilog3xi-(∑pilogxi)33(∑pilog2xi-(∑pilogxi)2)))1/s=lims→0(1+∑pilog3xi-(∑pilogxi)33(∑pilog2xi-(∑pilogxi)2)s(1+o(1)))1/s=exp(∑pilog3xi-(∑pilogxi)33(∑pilog2xi-(∑pilogxi)2)).Remark 1.1.
Analogously to the former considerations, one can define a class of Stolarsky means inn variables Er,s(x;n) as
(1.18)Er,s(x;n):=Wnr,ns(p0,x),
where p0={1/n}1n.
Therefore,(1.19)Er,s(x;n)=(r2s2∑1nxins-n∏1nxis∑1nxinr-n∏1nxir)1/n(s-r),rs(r-s)≠0.
Details are left to the readers.


## 1.1. Weighted Stolarsky Means

We introduce here a classW 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(1.9)W=Wr,s(p,q;x,y):=(r2s2(Ss)s-(G)s(Sr)r-(G)r)1/(s-r)=(r2s2pxs+qys-xpsyqspxr+qyr-xpryqr)1/(s-r).
Various properties concerning the means W can be established; some of them are the following:(1.10)Wr,s(p,q;x,y)=Ws,r(p,q;x,y);Wr,s(p,q;x,y)=Wr,s(q,p;y,x);Wr,s(p,q;y,x)=xyWr,s(p,q;x-1,y-1);War,as(p,q;x,y)=(Wr,s(p,q;xa,ya))1/a,a≠0.
Note that(1.11)W2r,2s(12,12;x,y)=(r2s2x2s+y2s-2(xy)2sx2r+y2r-2(xy)2r)1/2(s-r)=(r2s2(xs-ys)2(xr-yr)2)1/2(s-r)=E(r,s;x,y).In the same manner, we get(1.12)Wr,s(23,13;x3,y3)=(2xs+ys2xr+yr)1/(s-r)(E(r,s;x,y))2;Wr,s(34,14;x4,y4)=(3x2s-(xy)s+y2s3x2r-(xy)r+y2r)1/(s-r)(E(r,s;x,y))2.The weighted means from the classW can be extended continuously to the domain(1.13)D={(r,s;x,y)∣r,s∈ℝ;x,y∈ℝ+}.
This extension is given by(1.14)Wr,s(p,q;x,y)={(r2s2pxs+qys-xpsyqspxr+qyr-xpryqr)1/(s-r),rs(r-s)(x-y)≠0;(2pxs+qys-xpsyqspqs2log2(x/y))1/s,s(x-y)≠0,r=0;exp(-2s+pxslogx+qyslogy-(plogx+qlogy)xpsyqspxs+qys-xpsyqs),s(x-y)≠0,r=s;x(p+1)/3y(q+1)/3,x≠y,r=s=0;x,x=y.Note that those means are homogeneous of order1, that is, Wr,s(p,q;tx,ty)=tWr,s(p,q;x,y),t>0, symmetric in r,s, Wr,s(p,q;x,y)=Ws,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(1.15)Wr,s(p;x)={(r2(∑pixis-(∏xipi)s)s2(∑pixir-(∏xipi)r))1/(s-r),rs(s-r)≠0;(2s2∑pixis-(∏xipi)s∑pilog2xi-(∑pilogxi)2)1/s,r=0,s≠0;exp(-2s+∑pixislogxi-(∑pilogxi)(∏xipi)s∑pixis-(∏xipi)s),r=s≠0;exp(∑pilog3xi-(∑pilogxi)33(∑pilog2xi-(∑pilogxi)2)),r=s=0,
where x=(x1,x2,…,xn)∈ℝ+n,n≥2, p is an arbitrary positive weight sequence associated with x and Wr,s(p;x0)=a for x0=(a,a,…,a).We also write∑(·),∏(·) instead of ∑1n(·),∏1n(·).The above formulae are obtained by an appropriate limit process, implying continuity.For example, applying(1.16)ts=1+slogt+s22log2t+s36log3t+o(s3)(s→0),
we get(1.17)W0,0(p;x)=lims→0Ws,0(p;x)=lims→0(2s2∑pixis-(∏xipi)s∑pilog2xi-(∑pilogxi)2)1/s=lims→0(+(s36)log3(∏xipi))+o(s3))2s2(∑pilog2xi-(∑pilogxi)2)×((∑pi+s∑pilogxi+(s22)∑pilog2xi+(s36)∑pilog3xi)-(∑pi+slog(∏xipi)+(s22)log2(∏xipi)+(s36)log3(∏xipi))+o(s3))(∑pilog3xi-(∑pilogxi)33(∑pilog2xi-(∑pilogxi)2)))1/s=lims→0(1+∑pilog3xi-(∑pilogxi)33(∑pilog2xi-(∑pilogxi)2)s(1+o(1)))1/s=exp(∑pilog3xi-(∑pilogxi)33(∑pilog2xi-(∑pilogxi)2)).Remark 1.1.
Analogously to the former considerations, one can define a class of Stolarsky means inn variables Er,s(x;n) as
(1.18)Er,s(x;n):=Wnr,ns(p0,x),
where p0={1/n}1n.
Therefore,(1.19)Er,s(x;n)=(r2s2∑1nxins-n∏1nxis∑1nxinr-n∏1nxir)1/n(s-r),rs(r-s)≠0.
Details are left to the readers.


## 2. Results

The following basic assertion is of importance.Proposition 2.1.
The expressionsWr,s(p;x) are actual means, that is, for arbitrary weight sequence p one has
(2.1)min{x1,x2…,xn}≤Wr,s(p;x)≤max{x1,x2,…,xn}.Our main result is contained in the following.Proposition 2.2.
The meansWr,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 classW̅r,s(p,x).Assuming that all integrals exist,(2.2)W̅r,s(p,x)={(r2(∫p(t)xs(t)dt-exp(s∫p(t)logx(t)dt))s2(∫p(t)xr(t)dt-exp(r∫p(t)logx(t)dt)))1/(s-r),rs(s-r)≠0;(2s2∫p(t)xs(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)xs(t)logx(t)dt-(∫p(t)logx(t)dt)exp(s∫p(t)logx(t)dt)∫p(t)xs(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,
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.
W̅r,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 functionw(s), defined by
(3.1)w(s):={(12(πs)2(Γ(1+s)-e-γs))1/s,s≠0;exp(-γ-4ξ(3)π2),s=0,
is monotone increasing for s∈(-1,∞).
In particular, fors∈(-1,1), one has
(3.2)Γ(1-s)e-γs+Γ(1+s)eγs-πssin(πs)≤1-(πs)4144,
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 variableX and an arbitrary probability distribution with support on (-∞,+∞), it is well known that(3.3)EeX≥eEX.Denoting the central moment of orderk by μk=μk(X):=E(X-EX)k, we improve this inequality to the following propsositions.Proposition 3.2.
For an arbitrary probability law with support onℝ, one has
(3.4)EeX≥(1+(μ22)exp(μ33μ2))eEX.Proposition 3.3.
One also has that(3.5)(EesX-esEXs2σX2/2)1/s
is monotone increasing in s.


### 3.3. Shifted Stolarsky Means

Especially interesting is studying theshifted Stolarsky meansE*, defined by(3.6)Er,s*(x,y):=limp→0+Wr,s(p,q;x,y).
Their analytic continuation to the whole (r,s) plane is given by(3.7)Er,s*(x,y)={(r2(xs-ys(1+slog(x/y)))s2(xr-yr(1+rlog(x/y))))1/(s-r),rs(r-s)(x-y)≠0;(2s2xs-ys(1+slog(x/y))log2(x/y))1/s,s(x-y)≠0,r=0;exp(-2s+(xs-ys)logx-syslogylog(x/y)xs-ys(1+slog(x/y))),s(x-y)≠0,r=s;x1/3y2/3,r=s=0;x,x=y.Main results concerning the meansE* are contained in  the following propositions.Proposition 3.4.
MeansEr,s*(x,y) are monotone increasing in either r or s for each fixed x,y∈ℝ+.Proposition 3.5.
MeansEr,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 Er,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 intervalI,I⊂ℝ such that the means  Er,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 intervalI which includes zero and r,s∈I,
(i)  Er,s*(x,y) are not logarithmically convex (concave);
(ii)  Wr,s(p,q;x,y) are logarithmically convex (concave) if and only if p=q=1/2.


## 3.1. Applications in Analysis

As an illustration of the above, we give the following proposition.Proposition 3.1.
The functionw(s), defined by
(3.1)w(s):={(12(πs)2(Γ(1+s)-e-γs))1/s,s≠0;exp(-γ-4ξ(3)π2),s=0,
is monotone increasing for s∈(-1,∞).
In particular, fors∈(-1,1), one has
(3.2)Γ(1-s)e-γs+Γ(1+s)eγs-πssin(πs)≤1-(πs)4144,
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 variableX and an arbitrary probability distribution with support on (-∞,+∞), it is well known that(3.3)EeX≥eEX.Denoting the central moment of orderk by μk=μk(X):=E(X-EX)k, we improve this inequality to the following propsositions.Proposition 3.2.
For an arbitrary probability law with support onℝ, one has
(3.4)EeX≥(1+(μ22)exp(μ33μ2))eEX.Proposition 3.3.
One also has that(3.5)(EesX-esEXs2σX2/2)1/s
is monotone increasing in s.


## 3.3. Shifted Stolarsky Means

Especially interesting is studying theshifted Stolarsky meansE*, defined by(3.6)Er,s*(x,y):=limp→0+Wr,s(p,q;x,y).
Their analytic continuation to the whole (r,s) plane is given by(3.7)Er,s*(x,y)={(r2(xs-ys(1+slog(x/y)))s2(xr-yr(1+rlog(x/y))))1/(s-r),rs(r-s)(x-y)≠0;(2s2xs-ys(1+slog(x/y))log2(x/y))1/s,s(x-y)≠0,r=0;exp(-2s+(xs-ys)logx-syslogylog(x/y)xs-ys(1+slog(x/y))),s(x-y)≠0,r=s;x1/3y2/3,r=s=0;x,x=y.Main results concerning the meansE* are contained in  the following propositions.Proposition 3.4.
MeansEr,s*(x,y) are monotone increasing in either r or s for each fixed x,y∈ℝ+.Proposition 3.5.
MeansEr,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 Er,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 intervalI,I⊂ℝ such that the means  Er,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 intervalI which includes zero and r,s∈I,
(i)  Er,s*(x,y) are not logarithmically convex (concave);
(ii)  Wr,s(p,q;x,y) are logarithmically convex (concave) if and only if p=q=1/2.


## 4. Proofs

For the proof of Proposition2.1, we apply the following assertion on Jensen functionals Jf(p,x) from [6].Theorem 4.1.
Letf,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
(4.1)ϕ-1(Jn(p,x;f)Jn(p,x;g))(n≥2)
represents a mean value of the numbers x1,…,xn, that is,
(4.2)min{x1,…,xn}≤ϕ-1(Jn(p,x;f)Jn(p,x;g))≤max{x1,…,xn}
if and only if the relation
(4.3)f′′(t)=ϕ(t)g′′(t)
holds for each t∈I.Recall that the Jensen functionalJn(p,x;f) is defined on an interval I,I⊆ℝ by(4.4)Jn(p,x;f):=∑1npif(xi)-f(∑1npixi),
where f:I→ℝ,  x=(x1,x2,…,xn)∈In, and p={pi}1n is a positive weight sequence.The famous Jensen's inequality asserts that(4.5)Jn(p,x;f)≥0,
whenever f is a (strictly) convex function on I, with the equality case if and only if x1=x2=⋯=xn.Proof of Proposition2.1.
Define the auxiliary functionhs(x) by
(4.6)hs(x):={esx-sx-1s2,s≠0;x22,s=0.
Since
(4.7)hs′(x)={esx-1s,s≠0;x,s=0,hs′′(x)=esx,s∈ℝ,
we conclude that hs(x) is a continuously twice differentiable convex function on ℝ.
Denotingf(t):=hs(t),g(t):=hr(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 logWr,s(p,ex) represents a mean value, which is equivalent to the assertion of Proposition 2.1.Proof of Proposition2.2.
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.
Letfs(x) be a twice continuously differentiable function in x with a parameter s. If fs′′(x) is log-convex in s for s∈I:=(a,b);x∈K:=(c,d), then the Jensen functional
(4.8)Jf(w,x;s)=J(s):=∑wifs(xi)-fs(∑wixi),
is log-convex in s for s∈I,xi∈K,i=1,2,…, where w={wi} is any positive weight sequence.At the beginning, we need some preliminary lemmas.  Lemma 4.3.
A positive functionf is log-convex on I if and only if the relation
(4.9)f(s)u2+2f(s+t2)uw+f(t)w2≥0
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).
Ifg(x) is twice continuously differentiable and g′′(x)≥0 on K, then g(x) is convex on K and the inequality
(4.10)∑wig(xi)-g(∑wixi)≥0
holds for each xi∈K,i=1,2,…, and any positive weight sequence {wi},∑wi=1.Lemma 4.5.
For a convexf, the expression
(4.11)f(s)-f(r)s-r
is increasing in both variables.Proof of Theorem4.2.
Consider the functionF(x) defined as
(4.12)F(x)=F(u,v,s,t;x):=u2fs(x)+2uvf(s+t)/2(x)+v2ft(x),
where u,v∈ℝ;s,t∈I are real parameters independent of the variable x∈K.
Since(4.13)F′′(x)=u2fs′′(x)+2uvf(s+t)/2′′(x)+v2ft′′(x),
and by assuming fs′′(x) is log-convex in s, it follows from Lemma 4.3 that F′′(x)≥0,x∈K.
Therefore, by Lemma4.4, we get
(4.14)∑wiF(xi)-F(∑wixi)≥0,xi∈K,
which is equivalent to
(4.15)u2J(s)+2uvJ(s+t2)+v2J(t)≥0.
According to Lemma4.3 again, this is possible only if J(s) is log-convex and the proof is done.
Now, the proof of Proposition2.2 easily follows.
From the above, we see thaths(x) is twice continuously differentiable and that hs′′(x) is a log-convex function for each real s,x.
Applying Theorem4.2, we conclude that the form
(4.16)Φh(w,x;s)=Φ(s):={∑wiesxi-es∑wixis2,s≠0,∑wixi2-(∑wixi)22,s=0,
is log-convex in s.
By Lemma4.5, with f(s)=logΦ(s), we find out that
(4.17)logΦ(s)-logΦ(r)s-r=log(Φ(s)Φ(r))1/(s-r)
is monotone increasing either in s or r. Therefore, by changing variable xi→logxi, we finally obtain the proof of Proposition 2.2.Proof of Proposition2.3.
The assertion of Proposition2.3 follows from Proposition 2.2 by the standard argument (cf. [8, pages 131–134]). Details are left to the reader.Proof of Proposition3.1.
The proof follows puttingx(t)=t,p(t)=e-t,t∈(0,+∞) and applying Proposition 2.2. with r=0. Corresponding integrals are
(4.18)∫0∞e-tlogt=-γ;∫0∞e-tlog2t=γ2+π26;∫0∞e-tlog3t=-γ3-γπ22-2ξ(3),
with
(4.19)Γ(1-s)Γ(1+s)=πssin(πs).Proof of Proposition3.2.
By Proposition2.3, we get
(4.20)W0,1(p,ex)≥W0,0(p,ex),
that is,
(4.21)EeX-eEXμ2/2≥exp(EX3-(EX)33μ2).
Using the identity EX3-(EX)3=μ3+3μ2EX, we obtain the proof of Proposition 3.2.Proof of Proposition3.3.
This assertion is straightforward consequence of the fact thatW0,s(p,ex) is monotone increasing in s.Proof of Proposition3.4.
Direct consequence of Proposition2.2.Proof of Proposition3.5.
This is left as an easy exercise to the readers.Proof of Proposition3.6.
We prove only part (ii). The proof of (i) goes along the same lines.
Suppose that0∈(a,b):=I and that Er,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
(4.22)Fs(p,q;x,y):=W0,s(p,q;x,y)W0,-s(p,q;x,y)-(W0,0(p,q;x,y))2
is of constant sign for each x,y>0.
Substituting(x/y)s:=ew,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
(4.23)F(p,q;w):=pew+q-epw-e(2/3)(1+p)w(pe-w+q-e-pw).
Developing in power series in w, we get
(4.24)F(p,q;w)=11620pq(1+p)(2-p)(1-2p)w5+O(w6).
Therefore,F(p,q;w) can be of constant sign for each w∈ℝ only if p=1/2(=q).
Suppose now thatI 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
(4.25)W0,0(p,q;x,y)W0,2s(p,q;x,y)-(W0,s(p,q;x,y))2
is of constant sign for each x,y∈ℝ+.
Proceeding as before, this is equivalent to the assertion thatG(p,q;w) is of constant sign with
(4.26)G(p,q;w):=p3q3w6e(2/3)(p+1)w(pe2w+q-e2pw)-(pew+q-epw)4.
However,
(4.27)G(p,q;w)=2405p4q4(1+p)(1+q)(q-p)w11+O(w12).
Hence, we conclude thatG(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.



---

*Source: 432857-2009-10-29.xml*