IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation43285710.1155/2009/432857432857Research ArticleAn Extension of Stolarsky Means to the Multivariable CaseSimicSlavkoQiFengMathematical Institute SANUKneza Mihaila 3611000 BelgradeSerbiasanu.ac.rs200928092009200910072009230920092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

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

{(r,s;x,y)  r,s;x,y+} by the following:

Er,s(x,y)={(r(xs-ys)s(xr-yr))1/(s-r)rs(r-s)0;exp(-1s+xslogx-yslogyxs-ys),r=s0;(xs-yss(logx-logy))1/s,s0,r=0;xy,r=s=0;x,y=x>0, and in this form are introduced by Keneth Stolarsky in .

Most of the classical two-variable means are special cases of the class E. 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 class E to n,  n>2 variables. Unfortunately, this is done in a highly artificial mode (cf. ), without a practical background. Here is an illustration of this point; recently Merikowski  has proposed the following generalization of the Stolarsky mean Er,s to several variables:

Er,s(X):=[L(Xs)L(Xr)]1/(s-r),rs, where X=(x1,,xn) is an n-tuple of positive numbers and

L(Xs):=(n-1)!En-1i=1nxisuidu1dun-1.

The symbol En-1 stands for the Euclidean simplex which is defined by

En-1:={(u1,,un-1):ui0,1in-1;u1++un-11}.

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 Sr=Sr(p,q;x,y), defined by

G:=xpyq;Sr:=(pxr+qyr)1/r, where

p,q,x,y+;    p+q=1;r/{0}  . Note that (Sr)r>(G)r for xy,  r0, and limr0Sr=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,yR+,p+q=1,rs(r-s)(x-y)0, we define

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:

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,a0. Note that

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

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 class W can be extended continuously to the domain

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

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,xy,r=s=0;x,x=y.

Note that those means are homogeneous of order 1, 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

Wr,s(p;x)={(r2(pixis-(xipi)s)s2(pixir-(xipi)r))1/(s-r),rs(s-r)0;(2s2pixis-(xipi)spilog2xi-(pilogxi)2)1/s,r=0,s0;exp(-2s+pixislogxi-(pilogxi)(xipi)spixis-(xipi)s),r=s0;exp(pilog3xi-(pilogxi)33(pilog2xi-(pilogxi)2)),r=s=0, where x=(x1,x2,,xn)+n,  n2, 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

ts=1+slogt+s22log2t+s36log3t+o(s3)(s0), we get

W0,0(p;x)=lims0Ws,0(p;x)=lims0(2s2pixis-(xipi)spilog2xi-(pilogxi)2)1/s=lims0(+(s36)log3(xipi))+o(s3))2s2(pilog2xi-(pilogxi)2)×((pi+spilogxi+(s22)pilog2xi+(s36)pilog3xi)-(pi+slog(xipi)+(s22)log2(xipi)+(s36)log3(xipi))+o(s3))(pilog3xi-(pilogxi)33(pilog2xi-(pilogxi)2)))1/s=lims0(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 in n variables Er,s(x;n) as Er,s(x;n):=Wnr,ns(p0,x), where p0={1/n}1n.

Therefore, Er,s(x;n)=(r2s21nxins-n1nxis1nxinr-n1nxir)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 expressions Wr,s(p;x) are actual means, that is, for arbitrary weight sequence p one has min{x1,x2,xn}Wr,s(p;x)max{x1,x2,,xn}.

Our main result is contained in the following.

Proposition 2.2.

The means Wr,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 W̅r,s(p,x).

Assuming that all integrals exist,

W̅r,s(p,x)={(r2(p(t)xs(t)dt-exp(sp(t)logx(t)dt))s2(p(t)xr(t)dt-exp(rp(t)logx(t)dt)))1/(s-r),rs(s-r)0;(2s2p(t)xs(t)dt-exp(sp(t)logx(t)dt)p(t)log2x(t)dt-(p(t)logx(t)dt)2)1/s,r=0,s0;exp(-2s+p(t)xs(t)logx(t)dt-(p(t)logx(t)dt)exp(sp(t)logx(t)dt)p(t)xs(t)dt-exp(sp(t)logx(t)dt)),r=s0;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. Applications3.1. 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,s0;exp(-γ-4ξ(3)π2),s=0, is monotone increasing for s(-1,).

In particular, for s(-1,1), one has 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 variable X and an arbitrary probability distribution with support on (-,+), it is well known that

EeXeEX.

Denoting the central moment of order k 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 on , one has EeX(1+(μ22)exp(μ33μ2))eEX.

Proposition 3.3.

One also has that (EesX-esEXs2σX2/2  )1/s is monotone increasing in s.

3.3. Shifted Stolarsky Means

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

Er,s*(x,y):=limp0+Wr,s(p,q;x,y). Their analytic continuation to the whole (r,s) plane is given by

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 means E* are contained in the following propositions.

Proposition 3.4.

Means Er,s*(x,y) are monotone increasing in either r or s for each fixed x,y+.

Proposition 3.5.

Means Er,s*(x,y) are monotone increasing in either x or y for each r,s.

A well known result of Qi () 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 interval I,I such that the means Er,s*(x,y) are logarithmically convex (concave) for r,sI and each x,y+?

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

Proposition 3.6.

On any interval I which includes zero and r,sI,

(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 Proposition 2.1, we apply the following assertion on Jensen functionals Jf(p,x) from .

Theorem 4.1.

Let f,g:I 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(Jn(p,x;f)Jn(p,x;g))(n2) represents a mean value of the numbers x1,,xn, that is, min{x1,,xn}ϕ-1(Jn(p,x;f)Jn(p,x;g))max{x1,,xn} if and only if the relation f′′(t)=ϕ(t)g′′(t) holds for each tI.

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

Jn(p,x;f):=1npif(xi)-f(1npixi), where f:I, x=(x1,x2,,xn)In, and p={pi}1n is a positive weight sequence.

The famous Jensen's inequality asserts that

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 Proposition <xref ref-type="statement" rid="prop1">2.1</xref>.

Define the auxiliary function hs(x) by hs(x):={esx-sx-1s2,s0;x22,s=0. Since hs(x)={esx-1s,s0;x,s=0,hs(x)=esx,s, we conclude that hs(x) is a continuously twice differentiable convex function on .

Denoting f(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 Proposition <xref ref-type="statement" rid="prop2">2.2</xref>.

We prove first a global theorem concerning log-convexity of the Jensen's functional with a parameter, which can be very usable (cf. ).Theorem 4.2.

Let fs(x) be a twice continuously differentiable function in x with a parameter s. If fs(x) is log-convex in s for sI:=(a,b);    xK:=(c,d), then the Jensen functional Jf(w,x;s)=J(s):=wifs(xi)-fs(wixi), is log-convex in s for sI,  xiK,  i=1,2,, where w={wi} 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)w20 holds for each real u,w and s,tI.

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 wig(xi)-g(wixi)0 holds for each xiK,  i=1,2,, and any positive weight sequence {wi},  wi=1.

Lemma 4.5.

For a convex f, the expression f(s)-f(r)s-r is increasing in both variables.

Proof of Theorem <xref ref-type="statement" rid="thm2">4.2</xref>.

Consider the function F(x) defined as F(x)=F(u,v,s,t;x):=u2fs(x)+2uvf(s+t)/2(x)+v2ft(x), where u,v;    s,tI are real parameters independent of the variable xK.

Since 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,  xK.

Therefore, by Lemma 4.4, we get wiF(xi)-F(wixi)0,xiK, which is equivalent to u2J(s)+2uvJ(s+t2)+v2J(t)0.

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 hs(x) is twice continuously differentiable and that hs(x) is a log-convex function for each real s,x.

Applying Theorem 4.2, we conclude that the form Φh(w,x;s)=Φ(s):={wiesxi-eswixis2,s0,wixi2-(wixi)22,s=0, 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) is monotone increasing either in s or r. Therefore, by changing variable xilogxi, we finally obtain the proof of Proposition 2.2.

Proof of Proposition <xref ref-type="statement" rid="prop3">2.3</xref>.

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 <xref ref-type="statement" rid="prop4">3.1</xref>.

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), with Γ(1-s)Γ(1+s)=πssin(πs).

Proof of Proposition <xref ref-type="statement" rid="prop5">3.2</xref>.

By Proposition 2.3, we get W0,1(p,ex)W0,0(p,ex), that is, EeX-eEXμ2/2exp(EX3-(EX)33μ2). Using the identity EX3-(EX)3=μ3+3μ2EX, we obtain the proof of Proposition 3.2.

Proof of Proposition <xref ref-type="statement" rid="prop6">3.3</xref>.

This assertion is straightforward consequence of the fact that W0,s(p,ex) is monotone increasing in s.

Proof of Proposition <xref ref-type="statement" rid="prop7">3.4</xref>.

Direct consequence of Proposition 2.2.

Proof of Proposition <xref ref-type="statement" rid="prop8">3.5</xref>.

This is left as an easy exercise to the readers.

Proof of Proposition <xref ref-type="statement" rid="prop9">3.6</xref>.

We prove only part (ii). The proof of (i) goes along the same lines.

Suppose that 0(a,b):=I and that Er,s(p,q;x,y) are log-convex (concave) for r,sI and any fixed x,y+. Then there should be an s,  s>0 such that 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, 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):=pew+q-epw-e(2/3)(1+p)w(pe-w+q-e-pw). Developing in power series in w, we get 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 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,s0,sI such that 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+.

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(pe2w+q-e2pw)-(pew+q-epw)4. However, G(p,q;w)=2405p4q4(1+p)(1+q)(q-p)w11+O(w12).

Hence, we conclude that G(p,q;w) can be of constant sign for sufficiently small w,  w only if p=q=1/2. Combining this with Feng Qi theorem, the assertion from Proposition 3.6 follows.

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