# Necessary and Sufficient Conditions for Schur Geometrical Convexity of the Four-Parameter Homogeneous Means

Authors: Zhen-Hang Yang
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

The necessary and sufficient conditions for Schur geometrical convexity of the four-parameter means are given. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means.


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## Body

## 1. Introduction and Main Result

Letp,q∈ℝ and a,b>0. For a≠b the Stolarsky means are defined as(1.1)Sp,q(a,b)={(qpap-bpaq-bq)1/(p-q),pq(p-q)≠0,L1/p(ap,bp),p≠0,q=0,L1/q(aq,bq),q≠0,p=0,I1/p(ap,bp),p=q≠0,ab,p=q=0,
and Sp,q(a,a)=a (see [1]), where
(1.2)L(x,y)={x-ylnx-lny,x≠y,xx=y,(1.3)I(x,y)={(xxyy)1/(x-y),x≠y,x,x=y
are the logarithmic mean and identric (exponential) mean of positive numbers x and y, respectively.Another two-parameter family of means was introduced by Gini in [2]. That are defined as(1.4)Gp,q(a,b)={(ap+bpaq+bq)1/(p-q),p≠q,exp(aplna+bplnbap+bp),p=q.Stolarsky and Gini means both are contained in the so-called four-parameter means [3], which are defined as follows.Definition 1.1.
Let(a,b)∈ℝ+×ℝ+ with a≠b and (p,q),(r,s)∈ℝ×ℝ. Then the four-parameter homogeneous means denoted by F(p,q;r,s;a,b) are defined as follows:
(1.5)F(p,q;r,s;a,b)=(L(apr,bpr)L(aps,bps)L(aqs,bqs)L(aqr,bqr))1/(p-q)(r-s)ifpqrs(p-q)(r-s)≠0,
or
(1.6)F(p,q;r,s;a,b)=(apr-bpraps-bpsaqs-bqsaqr-bqr)1/(p-q)(r-s)ifpqrs(p-q)(r-s)≠0.
Ifpqrs(p-q)(r-s)=0, then F(p,q;r,s;a,b) are defined as their corresponding limits, for example:
(1.7)F(p,p;r,s;a,b)=limq→pF(p,q;r,s;a,b)=(I(apr,bpr)I(aps,bps))1/p(r-s),ifprs(r-s)≠0,p=q,F(p,0;r,s;a,b)=limq→0F(p,q;r,s;a,b)=(L(apr,bpr)L(aps,bps))1/p(r-s),ifprs(r-s)≠0,q=0,F(0,0;r,s;a,b)=limp→0F(p,0;r,s;a,b)=G(a,b),ifrs(r-s)≠0,p=q=0,
where L(x,y),I(x,y) denote logarithmic mean and identric (exponential) mean, respectively, G(a,b)=ab.The Schur convexity ofSp,q(a,b) and Gp,q(a,b) on (0,∞)×(0,∞) with respect to (a,b) was investigated by Qi et al. [4], Shi et al. [5], Li and Shi [6], and Chu and Zhang [7]. Until now, they have been perfectly solved by Chu and Zhang [7], Wang and Zhang [8], respectively. Recently, Chu and Xia also proved the same result as Wang and Zhang [9].The Schur convexity ofSp,q(a,b) and Gp,q(a,b) on [0,∞)×[0,∞) and (-∞,0]×(-∞,0] with respect to (p,q) was investigated by Qi [10] and Sándor [11], respectively. Now Schur convexity of a four-parameter homogeneous means family containing Stolarsky and Gini means on (-∞,∞)×(-∞,∞) with respect to (p,q) has been perfectly solved by Yang [12].The Schur geometrical convexity was introduced by Zhang [13]. In [8, 14], Wand and Zhang proved that Gp,q(a,b) is Schur geometrically convex (Schur geometrically concave) on (0,∞)×(0,∞) with respect to (a,b) if p+q≥(≤)0. Chu et al. [15] pointed out that this conclusion is also true for Sp,q(a,b). Shi et al. [5, 16], Li and Shi [6], and Gu and Shi [17] also obtained similar results.The purpose of this paper is to present the necessary and sufficient conditions for Schur geometrical convexity of the four-parameter homogeneous means. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means with respect to(a,b).Our main result is as follows.Theorem 1.2.
For fixed(p,q),(r,s)∈ℝ×ℝ the four-parameter homogeneous means F(p,q;r,s;a,b) are Schur geometrically convex (Schur geometrically concave) on (0,∞)×(0,∞) with respect to (a,b) if and only if (p+q)(r+s)>(<)0.


## 2. Definitions and Lemmas

Definition 2.1 (see [18, 19]).
Letx=(x1,x2,…,xn) and y=(y1,y2,…,yn)∈ℝn(n≥2).(i)
x is said to by majorized by y (in symbol x≺y) if
(2.1)∑i=1kx[i]≤∑i=1ky[i]for1≤k≤n-1,∑i=1nx[i]=∑i=1ny[i],
where x[1]≥x[2]⋯≥x[n] and y[1]≥y[2]⋯≥y[n] are rearrangements of x and y in a decreasing order.(ii)
x≥y means xi≥yi for all i=1,2,…,n. Let Ω⊂ℝn(n≥2). The function ϕ:Ω→ℝ is said to be increasing if x≥y implies ϕ(x)≥ϕ(y). ϕ is said to be decreasing if and only if -ϕis increasing.(iii)
Ω⊂ℝn is called a convex set if (αx1+βy1,…,αxn+βyn)∈Ω for all x and y, where α,β∈[0,1] with α+β=1.(iv)
LetΩ⊂ℝn(n≥2) be a set with nonempty interior. Then ϕ:Ω→ℝ is said to be Schur convex if x≺y on Ω implies ϕ(x)≤ϕ(y). ϕ is said to be Schur concave if -ϕ is Schur convex.Definition 2.2 (see [13, 20]).
Letx=(x1,x2,…,xn) and y=(y1,y2,…,yn)∈ℝ+n(n≥2). Denote
(2.2)lnx=(lnx1,lnx2,…,lnxn),lny=(lny1,lny2,…,lnyn).(i)
Ω⊂ℝ+n is called a geometrically convex set if (x1αy1β,…,xnαynβ)∈Ω for all x and y, where α,β∈[0,1] with α+β=1.(ii)
LetΩ⊂ℝ+n(n≥2) be a set with nonempty interior. Then function ϕ:Ω→ℝ+ is said to be Schur geometrically convex on Ω if lnx≺lny on Ω implies ϕ(x)≤ϕ(y). ϕ is said to be Schur geometrically concave if -ϕ is Schur geometrically convex.Definition 2.3 (see [18]).
(i)Ω⊂ℝn(n≥2) is called symmetric set if x∈Ω implies Px∈Ω for every n×n permutation matrix P.
(ii) The functionϕ:Ω→ℝ is called symmetric if for every permutation matrix P, ϕ(Px)=ϕ(x) for all x∈Ω.Lemma 2.4 (see [18, 19]).
LetΩ⊂ℝn be a symmetric set with nonempty interior Ω0 and ϕ:Ω→ℝ be continuous on Ω and differentiable in Ω0. Then ϕ is Schur convex (Schur concave) on Ω if and only if ϕ is symmetric on Ω and
(2.3)(x1-x2)(∂ϕ∂x1-∂ϕ∂x2)≥(≤)0
holds for any x=(x1,x2,…,xn)∈Ω0.Lemma 2.5 (see [13, Theorem 1.4, page 108]).
LetΩ⊂ℝ+n be a symmetric set with a nonempty interior geometrically convex set Ω0. Let ϕ:Ω→ℝ+ be continuous on Ω and differentiable in Ω0. Then ϕ is Schur geometrically convex (Schur geometrically concave) on Ω if and only if ϕ is symmetric on Ω and
(2.4)(lnx1-lnx2)(x1∂ϕ∂x1-x2∂ϕ∂x2)≥(≤)0
holds for any x=(x1,x2,…,xn)∈Ω0.


## 3. Schur Geometrical Convexity of Two-Parameter Homogeneous  Functions

The more general form of two-parameter homogeneous means is the so-called two-parameter homogenous functions first introduced by Yang [21]. For conveniences, we record it as follows.Definition 3.1.
Assume thatf: ℝ+×ℝ+→ℝ+∪{0} is n-order homogeneous, continuous and exists first partial derivatives and (a,b)∈ℝ+×ℝ+, (p,q)∈ℝ×ℝ.
Iff(x,y)>0for (x,y)∈ℝ+×ℝ+∖{(x,x):x∈ℝ+} and f(x,x)=0 for all x∈ℝ+, then define
(3.1)ℋf(p,q;a,b)=(f(ap,bp)f(aq,bq))1(p-q)ifp≠q,pq≠0,ℋf(p,p;a,b)=limq→pℋf(p,q;a,b)=Gf,p(a,b)ifp=q≠0,
where
(3.2)Gf,p(a,b)=Gf1/p(ap,bp),Gf(x,y)=exp(xfx(x,y)lnx+yfy(x,y)lnyf(x,y)),fx(x,y)and fy(x,y)denote first-order partial derivatives with respect to first and second component of f(x,y),respectively.
Iff(x,y)>0for all (x,y)∈ℝ+×ℝ+, then define further
(3.3)ℋf(p,0;a,b)=(f(ap,bp)f(1,1))1/pifp≠0,q=0;ℋf(0,q;a,b)=(f(aq,bq)f(1,1))1/qifp=0,q≠0;ℋf(0,0;a,b)=afx(1,1)/f(1,1)bfy(1,1)/f(1,1)ifp=q=0.
Since f(x,y) is a homogeneous function, ℋf(p,q;a,b) is also one and called a homogeneous function with parameters p and q and simply denoted by ℋf(p,q) or ℋf sometimes.Concerning the monotonicity and log-convexity of two-parameter homogeneous functions, there have been some literatures such as [3, 21, 22], which yield some new and interesting inequalities for means.The two-parameter homogeneous functionsℋf(p,q;a,b) have some well properties (see [21–23]) such as the following lemma.Lemma 3.2 (see [23]).
Letf:ℝ+×ℝ+→ℝ+ be a homogenous and differentiable function and
(3.4)T(t)=T(t;a,b):=lnf(at,bt),(t;a,b)∈ℝ×ℝ+×ℝ+.
Then we have
(3.5)∂T(t;a,b)∂t=atfx(at,bt)lna+btfy(at,bt)lnbf(at,bt),(3.6)lnℋf(p,q;a,b)=∫01∂T(tp+(1-t)q;a,b)∂tdt.Next we give another property.Lemma 3.3.
Letf:ℝ+×ℝ+→ℝ+ be a homogenous and m-time differentiable function. Then ℋf(p,q;a,b)∈Cm-1(ℝ×ℝ×ℝ+×ℝ+).Proof.
Sincef(x,y) has continuous partial derivatives of m order with respect to x,y on ℝ+×ℝ+, the integrand in (3.6) has continuous partial derivatives of m-1 order with respect to p,q,a,b on ℝ×ℝ×ℝ+×ℝ+, that is ℋf(p,q;a,b)∈Cm-1(ℝ×ℝ×ℝ+×ℝ+).For the Schur geometrical convexity, we have the following result.Theorem 3.4.
Assume thatf: ℝ+×ℝ+→ℝ+ is a symmetric, n-order homogeneous, continuous, and three-time differentiable function. If for any (x,y)∈ℝ+×ℝ+ with x≠y(3.7)𝒩(x,y)=(x-y)(x(lnf)x-y(lnf)y-2xyℐln(xy))>(<)0,whereℐ=(lnf)xy,
then ℋf(p,q;a,b) is Schur geometrically convex on (0,∞)×(0,∞) with respect to (a,b) if and only if p+q>(<)0 and Schur geometrically concave if and only if p+q<(>)0.Proof.
(1) In the case of p≠q. We have
(3.8)lnℋf(p,q;a,b)=lnf(ap,bp)-lnf(aq,bq)p-q.
Some simple partial derivative computations yield
(3.9)∂lnℋf∂a=1ℋf∂ℋf∂a=1p-q(pap-1fx(ap,bp)f(ap,bp)-qaq-1fx(aq,bq)f(aq,bq)),∂lnℋf∂b=1ℋf∂ℋf∂b=1p-q(pbp-1fy(ap,bp)f(ap,bp)-qbq-1fy(aq,bq)f(aq,bq)),
hence,
(3.10)1ℋf(a∂ℋf∂a-b∂ℋf∂b)=g(p)-g(q)p-q,
where
(3.11)g(t)=tatfx(at,bt)f(at,bt)-tbtfy(at,bt)f(at,bt).
It is easy to verify that g(t) is even on (-∞,∞). In fact, since f(x,y) is n-order homogeneous and symmetric, for arbitrary λ>0, we have
(3.12)f(λx,λy)=λnf(x,y),fx(λx,λy)=λn-1fx(x,y),fy(λx,λy)=λn-1fy(x,y),f(x,y)=f(y,x),fx(x,y)=fy(y,x),fy(x,y)=fx(y,x).
Thus,
(3.13)g(-t)=-ta-tfx(a-t,b-t)f(a-t,b-t)--tb-tfy(a-t,b-t)f(a-t,b-t)=-ta-t(atbt)-(n-1)fx(bt,at)(atbt)-nf(bt,at)--tbt(atbt)-(n-1)fy(bt,at)(atbt)-nf(bt,at)=-tbtfy(at,bt)f(at,bt)+tatfx(at,bt)f(at,bt)=g(t).
Let at=x, bt=y. Then
(3.14)g′(t)=x(lnf)x+t((xfx(x,y)f(x,y))xdxdt+(xfx(x,y)f(x,y))ydydt)-y(lnf)y-t((yfy(x,y)f(x,y))xdxdt+(yfy(x,y)f(x,y))ydydt)=x(lnf)x+t(x(xfx(x,y)f(x,y))xlna+y(xfx(x,y)f(x,y))ylnb)-y(lnf)y-t(x(yfy(x,y)f(x,y))xlna+y(yfy(x,y)f(x,y))ylnb).
Note xfx(x,y)/f(x,y) and yfy(x,y)/f(x,y) both are 0-order homogeneous with respect to x and y, then
(3.15)x(xfx(x,y)f(x,y))x+y(xfx(x,y)f(x,y))y=0,x(yfy(x,y)f(x,y))x+y(yfy(x,y)f(x,y))y=0,
and then
(3.16)x(xfx(x,y)f(x,y))x=-y(xfx(x,y)f(x,y))y=-xyℐ,y(yfy(x,y)f(x,y))y=-x(yfy(x,y)f(x,y))x=-xyℐ.
Therefore,
(3.17)g′(t)=x(lnf)x+txyℐ(lnb-lna)-y(lnf)y-txyℐ(lna-lnb)=x(lnf)x-y(lnf)y-2txyℐ(lna-lnb)=x(lnf)x-y(lnf)y-2xyℐln(xy)=𝒩(x,y)x-yforx≠y.
By the mean values theorem, there is a ξ between |p| and |q| such that
(3.18)g(p)-g(q)p-q=g(|p|)-g(|q|)p-q=|p|-|q|p-qg′(ξ)=p+q|p|+|q|g′(ξ)=p+q|p|+|q|𝒩(x,y)x-y,forx≠y,
where x=aξ, y=bξ. Thus we have
(3.19)(lna-lnb)(a∂ℋf∂a-b∂ℋf∂b)=ℋfp+q|p|+|q|ln(ab)𝒩(x,y)x-y=ℋfp+q|p|+|q|𝒩(x,y)ξlnx-lnyx-y={>0ifp+q>(<)0,<0ifp+q<(>)0.
By Lemma 2.5, our required result is derived immediately.
(2) In the case of p=q≠0. By Lemma 3.3 together with (3.10) and (3.17), we have
(3.20)1ℋf(p,p)(a∂ℋf(p,p)∂a-b∂ℋf(p,p)∂b)=limq→p1ℋf(p,q)(a∂ℋf(p,q)∂a-b∂ℋf(p,q)∂b)=limq→pg(p)-g(q)p-q=g′(p)=𝒩(x,y)x-y,
where x=ap, y=bp. Hence we have
(3.21)(lna-lnb)(a∂ℋf(p,p)∂a-b∂ℋf(p,p)∂b)=ℋf(p,p)(lna-lnb)𝒩(x,y)x-y=p-1ℋf(p,p)𝒩(x,y)lnx-lnyx-y={>0ifp>(<)0,<0ifp<(>)0.
By Lemma 2.5, the required result holds.
(3) In the case of p=q=0. By Lemma 3.3 and (3.20), we have
(3.22)1ℋf(0,0)(a∂ℋf(0,0)∂a-b∂ℋf(0,0)∂b)=limp→0(a∂ℋf(p,p)∂a-b∂ℋf(p,p)∂b)=limp→0g′(p).
However,
(3.23)g′(0)=(x(lnf)x-y(lnf)y-2xyℐln(xy))|x=1,y=1=1·fx(1,1)f(1,1)-1·fy(1,1)f(1,1)-2·1·1·ℐ(1,1)·ln(11)=0,
where fx(1,1)=fy(1,1) due to the symmetry of f(x,y). Thus
(3.24)(lna-lnb)(a∂ℋf(p,p)∂a-b∂ℋf(p,p)∂b)=0.
Summarizing the above three cases, this proof of Theorem3.4 is complete.


## 4. Proof of Main Result

Establishing the Theorem3.4, we are in a position to prove main result.Proof of Theorem1.2.
It follows from [3, Section 1], that F(p,q;r,s;a,b)=ℋℋL(p,q;a,b), where ℋL=ℋL(r,s)=ℋL(r,s;x,y)=Sr,s(x,y) is symmetric with respect to x and y. From Lemma 3.3, it follows that ℋL=ℋL(r,s;x,y)∈C∞(ℝ×ℝ×ℝ+×ℝ+). Thus we have
(4.1)(lnℋL(r,r))x=lims→r(lnℋL(r,s))x,(4.2)(lnℋL(r,r))y=lims→r(lnℋL(r,s))y,(4.3)(lnℋL(r,r))xy=lims→r(lnℋL(r,s))xy,(4.4)(lnℋL(r,0))x=lims→0(lnℋL(r,s))x,(4.5)(lnℋL(r,0))y=lims→0(lnℋL(r,s))y,(4.6)(lnℋL(r,0))xy=lims→r(lnℋL(r,s))xy,(4.7)(lnℋL(0,0))x=limr→0(lnℋL(r,r))x,(4.8)(lnℋL(0,0))y=limr→0(lnℋL(r,r))y,(4.9)(lnℋL(0,0))xy=limr→0(lnℋL(r,r))xy.(1) In the case of rs(r-s)≠0.
Simple partial derivative calculations yield(4.10)lnℋL=1r-s(ln|s|+ln|xr-yr|-ln|r|-ln|xs-ys|),(lnℋL)x=1r-s(rxr-1xr-yr-sxs-1xs-ys),(lnℋL)y=1r-s(-ryr-1xr-yr+sys-1xs-ys),ℐ=(lnℋL)xy=1xy(r-s)(r2xryr(xr-yr)2-s2xsys(xs-ys)2).
Hence,
(4.11)𝒩(x,y)=(x-y)(x(lnℋL)x-y(lnℋL)y-2xyℐln(xy))=x-yr-s(r(xr+yr)xr-yr-2r2xryrln(x/y)(xr-yr)2)-x-yr-s(s(xs+ys)xs-ys-2s2xsysln(x/y)(xs-ys)2)=(x-y)P(r)-P(s)r-s,
where
(4.12)P(t)=t(xt+ytxt-yt-2xtytln(xt/yt)(xt-yt)2).
It is easy to check that P(t) is even and increasing (decreasing) on (0,∞) if x>(<)y. Indeed,
(4.13)P(-t)=-t(x-t+y-tx-t-y-t-2x-ty-tln(x-t/y-t)(x-t-y-t)2)=P(t).
With (x/y)t=u, then t=lnu/ln(x/y), and then P(t) can be written as
(4.14)P(t)=1ln(x/y)(u+1u-1lnu-2uln2u(u-1)2).
Direct computation yields
(4.15)P′(t)=1ln(x/y)(u+1u-1lnu-2uln2u(u-1)2)′dudt=u((u+1)(u-1)/u-lnu(u-1)2+lnuu-1-2ln2u(u-1)2-4ulnuu-1(u-1)/u-lnu(u-1)2)(u-1)/lnu=L̲̲(u+1)L2-6uL+2u(u+1)(u-1)L2=2L(((u+1)/2)L-u)+4u((u+1)/2-L)(u-1)L2.
From
(4.16)u+12L-u=u2-1lnu2-u2>0,L-u+12<0,
it follows that P′(t)>0 if u-1>0, that is, x>y and P′(t)<0 if x<y. Namely,
(4.17)(x-y)P′(t)>0fort>0withx≠y.
By the mean values theorem, there is aη between |r| and |s| such that
(4.18)P(|r|)-P(|s|)=(|r|-|s|)P′(η),
and then
(4.19)𝒩(x,y)=(x-y)P(r)-P(s)r-s=(x-y)r+s|r|+|s|P(|r|)-P(|s|)|r|-|s|=r+s|r|+|s|·(x-y)P′(η)={>0ifr+s>0,<0ifr+s<0.
Using Theorem 3.4, for fixed (p,q),(r,s)∈ℝ×ℝ with rs(r-s)≠0, the four-parameter homogeneous means F(p,q;r,s;a,b) are Schur geometrically convex on (0,∞)×(0,∞) with respect to (a,b) if and only if (p+q)(r+s)>0 and Schur geometrically concave if and only if (p+q)(r+s)<0.(2) In the case of s=0, r≠0.
From (4.11) together with (4.4)–(4.6) and (4.19), there is a η1 between 0 and |r| such that
(4.20)𝒩(x,y)=(x-y)(x(lnℋL(r,0))x-y(lnℋL(r,0))y-2xy(lnℋL(r,0))xyln(xy))=lims→0((x-y)(x(lnℋL(r,s))x-y(lnℋL(r,s))y-2xy(lnℋL(r,s))xyln(xy)))=lims→0(x-y)P(r)-P(s)r-s=lims→0r+s|r|+|s|·lims→0(x-y)P′(η1)={>0ifr>0,<0ifr<0,(by(4.17)).(3) In the case of r=0, s≠0.
SinceℋL(r,s;x,y) is symmetric with respect to r and s, it follows from case 2 that
(4.21)𝒩(x,y)=(x-y)(x(lnℋL(0,s))x-y(lnℋL(0,s))y-2xy(lnℋL(r,s))xyln(xy))={>0ifs>0,<0ifs<0.(4) In the case of r=s≠0.
From (4.11) together with (4.1)–(4.3), we have (4.22)𝒩(x,y)=(x-y)(x(lnℋL(r,r))x-y(lnℋL(r,r))y-2xy(lnℋL(r,r))xyln(xy))=lims→r((x-y)(x(lnℋL(r,s))x-y(lnℋL(r,s))y-2xy(lnℋL(r,s))xyln(xy)))=(x-y)lims→rP(r)-P(s)r-s=(x-y)P′(r)={>0ifr>0,<0ifr<0.(by(4.17))(5) In the case of r=s=0.
From (4.22) together with (4.7)–(4.9), we have
(4.23)𝒩(x,y)=(x-y)(x(lnℋL(0,0))x-y(lnℋL(0,0))y-2xy(lnℋL(0,0))xyln(xy))=limr→0((x-y)(x(lnℋL(r,r))x-y(lnℋL(r,r))y-2xy(lnℋL(r,r))xyln(xy)))=(x-y)limr→0P′(r).
But by (4.15) and some limit computations, we obtain
(4.24)limt→0P′(t)(x/y)t=u̲̲limu→1u((u+1)(u-1)/u-lnu(u-1)2+lnuu-1-2ln2u(u-1)2-4ulnuu-1(u-1)/u-lnu(u-1)2)=0,
which implies 𝒩(x,y)=0.
Summarizing the above five cases, our required results are derived.
This proof ends.


## 5. Other Corollaries

The four-parameter homogeneous meansF(p,q;r,s;a,b) also contain many other two-parameter means, for instance, for the identric (exponential) mean defined by (1.3), its two-parameter means are defined as follows [21, Example 2.3]: (5.1)ℋI(p,q;a,b)={(I(ap,bp)I(aq,bq))1/(p-q),p≠q,pq≠0,GI,p(a,b),p=q≠0,I1/p(ap,bp),p≠0,q=0,I1/q(aq,bq),p=0,q≠0,G(a,b),p=q=0,
where GI,p(a,b)=Y1/p(ap,bp):=Yp(a,b),Y(a,b)=Ie1-G2/L2.By [3], we see that (5.2)ℋI(p,q;a,b)=F(p,q;1,1;a,b).
And then according to Theorem 1.2, we have the following corollary.Corollary 5.1.
For fixed(p,q)∈ℝ×ℝ, the two-parameter identric (exponential) means ℋI(p,q;a,b) are Schur geometrically convex on (0,∞)×(0,∞) with respect to (a,b) if and only if p+q>0 and Schur geometrically concave if and only if p+q<0.As another example, for Heronian mean defined by(5.3)He=a+ab+b3,
its two-parameter means are defined as follows:(5.4)ℋHe(p,q;a,b)={(ap+(ab)p+bpaq+(ab)q+bq)1/(p-q),p≠q,pq≠0,a(ap+(ab)p/2)/(ap+(ab)p+bp)b((ab)p/2+bp)/(ap+(ab)p+bp),p=q≠0,He1/p(ap,bp),p≠0,q=0,He1/q(aq,bq),p=0,q≠0,G(a,b),p=q=0.By [3], we see that (5.5)ℋHe(p,q;a,b)=F(p,q;3/2,1/2;a,b).
And then according to Theorem 1.2, we have the following corollary.Corollary 5.2.
For fixed(p,q)∈ℝ×ℝ, the two-parameter Heronian means ℋHe(p,q;a,b) are Schur geometrically convex on (0,∞)×(0,∞) with respect to (a,b) if and only if p+q>0 and Schur geometrically concave if and only if p+q<0.



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*Source: 830163-2010-04-22.xml*