# The Schur-Convexity of the Generalized  Muirhead-Heronian Means

Authors: Yong-Ping Deng; Shan-He Wu; Yu-Ming Chu; Deng He
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We give a unified generalization of the generalized Muirhead means and the generalized Heronian means involving three parameters. The Schur-convexity of the generalized Muirhead-Heronian means is investigated. Our main result implies the sufficient conditions of the Schur-convexity of the generalized Heronian means and the generalized Muirhead means.


---

## Body

## 1. Introduction

In what follows, we denote the set of real numbers byR, the set of nonnegative real numbers by R+, and the set of positive real numbers by R++.Let(x,y)∈R++2; the classical Heronian means is defined by (see [1])
(1)He(x,y)=x+xy+y3.In 1999, Mao [2] gave the definition of dual Heronian means; that is,
(2)H~e(x,y)=x+4xy+y6.In 2001, Janous [3] considered the unified generalization of Heronian means He(x,y) and H~e(x,y) and presented a weighted generalization of the above-mentioned Heronian-type means, as follows:
(3)Hw(x,y)={x+wxy+yw+2,0≤w<∞,xy,w=∞.Jia and Cao [4] investigated the exponential generalization of Heronian means
(4)Hp(x,y)={(xp+(xy)p/2+yp3)1/p,p≠0,xy,p=0,
and they established some related inequalities. The monotonicity and Schur-convexity of the Heronian means Hp(x,y) were discussed by Li et al. in [5].Shi et al. [6] discussed the Schur-convexity of a further generalization of the Heronian means given by
(5)Hp,w(x,y)={(xp+w(xy)p/2+ypw+2)1/p,p≠0,xy,p=0,
and they obtained a significant result asserted by Theorem A below.Theorem A.
For fixed(p,w)∈R2,(1)
if(p,w)∈{(p,w)∣p≥2,0≤w≤2}, then Hp,w(x,y) is Schur-convex for (x,y)∈R+2.(2)
If(p,w)∈{(p,w)∣p≤1,w≥0}∪{(p,w)∣1<p≤3/2,w≥1}∪{(p,w)∣3/2<p≤2,w≥2}, then Hp,w(x,y) is Schur-concave for (x,y)∈R+2.As a further investigation of Theorem A, Fu et al. [7] gave the necessary and sufficient condition for the Schur-convexity of the generalized Heronian means Hp,w(x,y), which is stated in the following theorem.Theorem B.
For fixed(p,w)∈R2, the generalized Heronian means Hp,w(x,y) is Schur-convex for (x,y)∈R++2 if and only if
(6)(p,w)∈{(p,w)∣p≥2,0≤w≤2(p-1)}∪{(p,w)∣1<p≤2,w=0}.
Furthermore,Hp,w(x,y) is Schur-concave for (x,y)∈R++2 if and only if
(7)(p,w)∈{(p,w)∣p≤2,max⁡⁡{0,2(p-1)}≤w}.Remark 1.
It is easy to observe that, forp=1, w=0, H1,0(x,y)=(x+y)/2 is Schur-convex and Schur-concave for (x,y)∈R++2. In addition, we note that {(p,w)∣p=2,w=0}⊂{(p,w)∣p≥2,0≤w≤2(p-1)}. Thus, the conditions of Schur-convexity of Hp,w(x,y) in Theorem B can be rewritten as
(8)(p,w)∈{(p,w)∣p≥2,0≤w≤2(p-1)}∪{(p,w)∣1≤p<2,w=0}.
The Schur-power-convexity ofHp,w(x,y) was investigated by Yang [8].
In 2006, Trif [9] considered the following generalized Muirhead means, defined by
(9)M(p,q;x,y)=(xpyq+xqyp2)1/(p+q),
where x,y∈R++, p,q∈R, p+q≠0.
Gong et al. [10] investigated the Schur-convexity of generalized Muirhead means M(p,q;x,y) and obtained the following results.Theorem C.
For fixed(p,q)∈R2, the generalized Muirhead means M(p,q;x,y) is Schur-convex for (x,y)∈R++2 if and only if
(10)(p,q)∈{(p,q)∣(p-q)2≥p+q>0,pq≤0}.
Furthermore,M(p,q;x,y) is Schur-concave for (x,y)∈R++2 if and only if
(11)(p,q)∈{(p,q)∣p≥0,q≥0,(p-q)2≤p+q,nnn(p,q)≠(0,0)}∪{(p,q)∣p+q<0(p-q)2(p-q)2}.Remark 2.
If we define, forp=0, q=0, the generalized Muirhead means by M(0,0;x,y)=xy, we can easily find that M(0,0;x,y) is Schur-concave for (x,y)∈R++2; thereby, the conditions of Schur-concave of M(p,q;x,y) in Theorem C can be rewritten as
(12)(p,q)∈{(p,q)∣p≥0,q≥0,(p-q)2≤p+q}∪{(p,q)∣p+q<0}.
The Schur-geometric-convexity and Schur-harmonic-convexity of the generalized Muirhead meansM(p,q;x,y) were studied by Xia and Chu in [11, 12].
In this paper we generalize the generalized Muirhead meansM(p,q;x,y) and the generalized Heronian means Hp,w(x,y) in a unified form. For this purpose we define a generalized Muirhead-Heronian means Hp,q,w(x,y), as follows:
(13)Hp,q,w(x,y)={(xpyq+w(xy)(p+q)/2+xqyp2+w)1/(p+q),p+q≠0,xy,p=q=0,
where (p,q)∈R2, (x,y)∈R++2.The paper is organized as follows. Section2 introduces several definitions and lemmas; Section 3 discusses the Schur-convexity of the generalized Muirhead-Heronian means; Section 4 provides some remarks on the results given in Theorem 9 and it is shown that the sufficient conditions of Schur-convexity of the generalized Heronian means Hp,w(x,y) and the generalized Muirhead means M(p,q;x,y) can be deduced from Theorem 9 as special cases.


## 2. Definitions and Lemmas

We introduce and establish several definitions and lemmas, which will be used in the proofs of the main results in Section3.Definition 3 (see [13, page 7]).
For anyx=(x1,x2,…,xn), y=(y1,y2,…,yn)∈Rn, let x[1]≥x[2]≥⋯≥x[n] and y[1]≥y[2]≥⋯≥y[n] denote the components of x and y in decreasing order, respectively.
Then-tuple y is said to majorize x (or x is to be majorized by y), in symbols x≺y, if
(14)∑i=1k‍x[i]≤∑i=1k‍y[i]holdsfork=1,2,…,n-1,∑i=1n‍xi=∑i=1n‍yi.Definition 4 (see [13, page 54]).
For anyx=(x1,x2,…,xn), y=(y1,y2,…,yn)∈Ω (Ω⊂Rn), ϕ:Ω→R is said to be a Schur-convex function on Ω if x≺y on Ω implies ϕ(x)≤ϕ(y) and ϕ is said to be a Schur-concave function on Ω if and only if -ϕ is a Schur-convex function.Lemma 5 (see [13, page 57]).
LetΩ(⊂Rn) be a symmetric convex set with nonempty interior Ωo and ϕ:Ω→R a continuous symmetric function on Ω. If ϕ is differentiable on Ωo, then ϕ is the Schur-convex (Schur-concave) function on Ω if and only if
(15)(x1-x2)(∂ϕ∂x1-∂ϕ∂x2)≥0(≤0)
holds for all (x1,x2,…,xn)∈Ωo.Lemma 6.
Suppose thatp,q∈R, p>q, λ≥1 and
(16)g(λ)=pλp-q+q+w(p+q)2λ(p-q)/2-qλp-q+1-pλ-w(p+q)2λ((p-q)/2)+1.
Suppose also that
(17)B1={(p,q,w)∣p+q>0,p-q-2≥0,w≥0},B2={(p,q,w)∣p+q>0,p-q-2<0,w≥0},E11=B1∩{(p-q)2(p,q,w)∣q<0,nnn2(p-q)2-(2+w)(p+q)≥0},E12=B2∩{(p,q,w)∣q<0,nnn2(p-q)2-(2+w)(p+q)≥0,nnn(p+q)(p-q-2)wnnn-8q(p-q+1)≥0}.
Then(p+q)g(λ)≥0 for (p,q,w)∈E11∪E12.Proof.
Differentiatingg(λ) with respect to λ gives
(18)g′(λ)=p(p-q)λp-q-1+w(p+q)(p-q)4λ((p-q)/2)-1-q(p-q+1)λp-q-p-w(p+q)(p-q+2)4λ(p-q)/2,g′(1)=2(p-q)2-(2+w)(p+q)2.
Letf(λ)=λq-p+2g′′(λ), then
(19)f(λ)=p(p-q)(p-q-1)+w(p+q)(p-q)(p-q-2)8λ(q-p)/2-q(p-q+1)(p-q)λ-w(p+q)(p-q+2)(p-q)8λ((q-p)/2)+1,f(1)=p-q2(2(p-q)2-(2+w)(p+q)).
Differentiatingf(λ) with respect to λ yields
(20)f′(λ)=-w(p+q)(p-q)2(p-q-2)16λ((q-p)/2)-1-q(p-q+1)(p-q)+w(p+q)(p-q+2)(p-q)(p-q-2)16λ(q-p)/2,f′(1)=p-q8((p+q)(p-q-2)w-8q(p-q+1)),f′′(λ)=-w(p+q)(p-q)2(p-q-2)(p-q+2)32×(λ-1)λ((q-p)/2)-2.
In order to prove Lemma6, we need to consider the two cases below.
Case  1 (
(
p
,
q
,
w
)
∈
E
11
). In view of (p,q,w)∈E11, we have f′′(λ)≤0 for λ≥1. Hence f′(λ) is decreasing on [1,+∞), by which, together with
(21)f′(1)>0,lim⁡λ→+∞f′(λ)=-q(p-q+1)(p-q)>0,
we deduce that f′(λ)>0 for λ≥1. This means that f(λ) is increasing on [1,+∞). Thus, we have, for λ≥1,
(22)f(λ)≥f(1)≥0⟹g′′(λ)=f(λ)λq-p+2≥0⟹g′(λ)≥g′(1)≥0⟹g(λ)≥g(1)=0.
This leads to(p+q)g(λ)≥0.
Case  2 (
(
p
,
q
,
w
)
∈
E
12
). By (p,q,w)∈E12, we have f′′(λ)≥0 for λ≥1. Thus f′(λ) is increasing on [1,+∞), by which, together with
(23)f′(1)>0,lim⁡λ→+∞f′(λ)=-q(p-q+1)(p-q)>0,
we obtain f′(λ)>0 for λ≥1. It follows that f(λ) is increasing on [1,+∞). Thus, we have, for λ≥1,
(24)f(λ)≥f(1)≥0⟹g′′(λ)=f(λ)λq-p+2≥0⟹g′(λ)≥g′(1)≥0⟹g(λ)≥g(1)=0.
This implies that(p+q)g(λ)≥0. The proof of Lemma 6 is complete.Lemma 7.
Suppose thatp,q∈R, p>q, λ≥1 and
(25)g(λ)=pλp-q+q+w(p+q)2λ(p-q)/2-qλp-q+1-pλ-w(p+q)2λ((p-q)/2)+1.
Suppose also that
(26)B1={(p,q,w)∣p+q>0,p-q-2≥0,w≥0},B2={(p,q,w)∣p+q>0,p-q-2<0,w≥0},B3={(p,q,w)∣p+q<0,p-q-2≥0,w≥0},B4={(p,q,w)∣p+q<0,p-q-2<0,w≥0},E21=B1∩{(p,q,w)∣q>0,nnnnnnn2(p-q)2-(2+w)(p+q)≤0,nnnnnnn(p+q)(p-q-2)wnnnnnnn-8q(p-q+1)≤0},E22=B2∩{(p-q)2(p,q,w)∣q>0,nnnnnnn2(p-q)2-(2+w)(p+q)≤0},E23=B3∩{(p,q,w)∣q<0,nnnnnnn2(p-q)2-(2+w)(p+q)>0,nnnnnnn(p+q)(p-q-2)wnnnnnnn-8q(p-q+1)≥0},E24=B4∩{(p-q)2(p,q,w)∣q<0,nnnnnnnnnn2(p-q)2-(2+w)(p+q)>0}.
Then(p+q)g(λ)≤0 for (p,q,w)∈E21∪E22∪E23∪E24.Proof.
Using the differential expressions obtained in Lemma6, one has
(27)g′(λ)=p(p-q)λp-q-1+w(p+q)(p-q)4λ((p-q)/2)-1-q(p-q+1)λp-q-p-w(p+q)(p-q+2)4λ(p-q)/2,g′(1)=2(p-q)2-(2+w)(p+q)2,f(λ)=λq-p+2g′′(λ),f(λ)=p(p-q)(p-q-1)+w(p+q)(p-q)(p-q-2)8λ(q-p)/2-q(p-q+1)(p-q)λ-w(p+q)(p-q+2)(p-q)8λ((q-p)/2)+1,f(1)=p-q2(2(p-q)2-(2+w)(p+q)),f′(λ)=-w(p+q)(p-q)2(p-q-2)16λ((q-p)/2)-1-q(p-q+1)(p-q)+w(p+q)(p-q+2)(p-q)(p-q-2)16λ(q-p)/2,f′(1)=p-q8((p+q)(p-q-2)w-8q(p-q+1)),f′′(λ)=-w(p+q)(p-q)2(p-q-2)(p-q+2)32×(λ-1)λ((q-p)/2)-2.
We divide the proof of Lemma7 into four cases.
Case  1. If (p,q,w)∈E21, then
(28)f′(1)≤0,f(1)≤0,g′(1)≤0,f′′(λ)≤0.
Thus we have, forλ∈[1,+∞),
(29)f′′(λ)≤0⟹f′(λ)isdecreasing⟹f′(λ)≤0⟹f(λ)isdecreasing⟹f(λ)≤0⟹g′′(λ)≤0⟹g′(λ)isdecreasing⟹g′(λ)≤0⟹g(λ)isdecreasing⟹g(λ)≤g(1)=0⟹(p+q)g(λ)≤0.Case  2. If (p,q,w)∈E22, then
(30)f′(1)<0,lim⁡λ→+∞f′(λ)=-q(p-q+1)(p-q)<0,f(1)≤0,g′(1)≤0,f′′(λ)≥0.
Thus we have, forλ∈[1,+∞),
(31)f′′(λ)≥0⟹f′(λ)isincreasing⟹f′(λ)≤0⟹f(λ)isdecreasing⟹f(λ)≤0⟹g′′(λ)≤0⟹g′(λ)isdecreasing⟹g′(λ)≤0⟹g(λ)isdecreasing⟹g(λ)≤g(1)=0⟹(p+q)g(λ)≤0.Case  3. If (p,q,w)∈E23, then
(32)f′(1)≥0,f(1)>0,g′(1)>0,f′′(λ)≥0.
Thus we have, forλ∈[1,+∞),
(33)f′′(λ)≥0⟹f′(λ)isincreasing⟹f′(λ)≥0⟹f(λ)isincreasing⟹f(λ)>0⟹g′′(λ)>0⟹g′(λ)isincreasing⟹g′(λ)>0⟹g(λ)isincreasing⟹g(λ)≥g(1)=0⟹(p+q)g(λ)≤0.Case  4. If (p,q,w)∈E24, then
(34)f′(1)>0,f(1)>0,lim⁡λ→+∞f′(λ)=-q(p-q+1)(p-q)>0,g′(1)>0,f′′(λ)<0.
Thus we have, forλ∈[1,+∞),
(35)f′′(λ)<0⟹f′(λ)isdecreasing⟹f′(λ)>0⟹f(λ)isincreasing⟹f(λ)>0⟹g′′(λ)>0⟹g′(λ)isincreasing⟹g′(λ)>0⟹g(λ)isincreasing⟹g(λ)≥g(1)=0⟹(p+q)g(λ)≤0.
This completes the proof of Lemma 7.Lemma 8.
Suppose thatp,q∈R, p>q, λ≥1 and
(36)g(λ)=pλp-q+q+w(p+q)2λ(p-q)/2-qλp-q+1-pλ-w(p+q)2λ((p-q)/2)+1.
Suppose also that
(37)B1={(p,q,w)∣p+q>0,p-q-2≥0,w≥0},B3={(p,q,w)∣p+q<0,p-q-2≥0,w≥0},E31=B1∩{(p-q)2(p,q,w)∣q>0,NNNNN(p+q)(p-q-2)w-8q(p-q+1)>0,NNNNN2(p-q)2-(2+w)(p+q)<0}∩{(p,q,w)∣(p-q)2-3(p+q)+2≤0},E32=B3∩{(p,q,w)∣q<0,(p+q)(p-q-2)wnnnnnn-8q(p-q+1)<0}.
Then(p+q)g(λ)≤0 for (p,q,w)∈E31∪E32.Proof.
Based on the differential expressionsg′(λ), g′′(λ), g′(1), f′(λ), f′′(λ), f′(1) obtained in the proof of Lemma 6, in order to prove Lemma 8, we need to consider the two cases below.
Case  1. If (p,q,w)∈E31, then
(38)f′′(λ)≤0,f′(1)>0,lim⁡λ→+∞f′(λ)=-q(p-q+1)(p-q)<0.
Hence, we deduce that there existsλ1∈(1,+∞) such that f′(λ1)=0, satisfying f′(λ)>0 for λ∈[1,λ1) and f′(λ)<0 for λ∈(λ1,+∞).
Further, we conclude thatf(λ) is increasing on [1,λ1) and decreasing on (λ1,+∞); thereby, we get f(λ)≤f(λ1) for λ∈[1,+∞).
Fromf′(λ1)=0 we have
(39)(w(p+q)(p-q)2(p-q-2)16λ1w(p+q)(p-q+2)(p-q)(p-q-2)16nn-w(p+q)(p-q)2(p-q-2)16λ1)λ1(q-p)/2=q(p-q+1)(p-q);
this yields
(40)λ1(q-p)/2=16q(p-q+1)λ1(p+q)((p-q+2)λ1+q-p)(p-q-2)w;
we thus have
(41)f(λ1)=p(p-q)(p-q-1)-q(p-q+1)(p-q)λ1+(w(p+q)(p-q)(p-q-2)8nnnnn-w(p+q)(p-q+2)(p-q)λ18)λ1(q-p)/2=(p-q)2G1(λ1)(p-q-2)((p-q+2)λ1+q-p),
where
(42)G1(λ1)=-q(p-q+2)(p-q+1)(λ1-1)2+((p-q)2-3(p+q)+2)×(2+(p-q+2)(λ1-1))<0.
Note that(p,q,w)∈E31 implies p-q-2>0 and
(43)(p-q+2)λ1+q-p>(p-q+2)+q-p=2;
we conclude that f(λ)≤f(λ1)<0 for λ∈[1,+∞).
Hence, fromg′(1)<0, one has, for λ∈[1,+∞),
(44)f(λ)<0⟹g′′(λ)<0⟹g′(λ)isdecreasing⟹g′(λ)<0⟹g(λ)isdecreasing⟹g(λ)≤g(1)=0⟹(p+q)g(λ)≤0.
Case  2. If (p,q,w)∈E32, then
(45)f′′(λ)≥0,f′(1)<0,lim⁡λ→+∞f′(λ)=-q(p-q+1)(p-q)>0.
Thus, we deduce that there existsλ2∈(1,+∞) such that f′(λ2)=0, satisfying f′(λ)<0 for λ∈[1,λ2) and f′(λ)>0 for λ∈(λ2,+∞).
It follows thatf(λ) is decreasing on [1,λ2) and increasing on (λ2,+∞), therefore, we obtain
(46)f(λ)≥f(λ2)forλ∈[1,+∞).
Fromf′(λ2)=0, we have
(47)(w(p+q)(p-q)2(p-q-2)16λ2w(p+q)(p-q+2)(p-q)(p-q-2)16-w(p+q)(p-q)2(p-q-2)16λ2)λ2(q-p)/2=q(p-q+1)(p-q);
that is,
(48)λ2(q-p)/2=16q(p-q+1)λ2(p+q)((p-q+2)λ2+q-p)(p-q-2)w;
we thus have
(49)f(λ2)=p(p-q)(p-q-1)-q(p-q+1)(p-q)λ2+(w(p+q)(p-q)(p-q-2)8nnnnn-w(p+q)(p-q+2)(p-q)λ28)λ2(q-p)/2=(p-q)2G2(λ2)(p-q-2)((p-q+2)λ2+q-p),
where
(50)G2(λ2)=-q(p-q+2)(p-q+1)(λ2-1)2+((p-q)2-3(p+q)+2)×(2+(p-q+2)(λ2-1))>0.
Note that(p,q,w)∈E32 implies p-q-2>0 and
(51)(p-q+2)λ2+q-p>(p-q+2)+q-p=2,
which yields thatf(λ)≥f(λ2)>0 for λ∈[1,+∞).
Therefore, fromg′(1)>0, one has, for λ∈[1,+∞),
(52)f(λ)>0⟹g′′(λ)>0⟹g′(λ)isincreasing⟹g′(λ)>0⟹g(λ)isincreasing⟹g(λ)≥g(1)=0⟹(p+q)g(λ)≤0.
The proof of Lemma8 is completed.


## 3. Main Result

The main result of this paper is given by Theorem9 below.Theorem 9.
For fixed(p,q,w)∈R3, let
(53)A1={(p,q,w)∣p-q-2≥0,q≤0}∪{(p,q,w)∣p-q-2<0,q≤0,nnn(p+q)(p-q-2)wnnn-8q(p-q+1)≥0},A2={(p,q,w)∣q-p-2≥0,p≤0}∪{(p,q,w)∣q-p-2<0,p≤0,nnn(p+q)(q-p-2)wnnn-8p(q-p+1)≥0},A3={(p,q,w)∣p-q-2≥0,q>0,nnn(p+q)(p-q-2)wnnn-8q(p-q+1)≤0}∪{(p,q,w)∣p>q,p-q-2<0,q>0},A4={(p,q,w)∣q-p-2≥0,p>0,nnn(p+q)(q-p-2)wnnn-8p(q-p+1)≤0}∪{(p,q,w)∣q>p,q-p-2<0,p>0},A5={(p,q,w)∣p-q-2≥0,q<0,nnn(p+q)(p-q-2)wnnn-8q(p-q+1)≥0}∪{(p,q,w)∣p>q,p-q-2<0,q<0},A6={(p,q,w)∣q-p-2≥0,p<0,nnn(p+q)(q-p-2)wnnn-8p(q-p+1)≥0}∪{(p,q,w)∣q>p,q-p-2<0,p<0},A7={(p,q,w)∣p-q-2≥0,q>0,nnn(p+q)(p-q-2)wnnn-8q(p-q+1)>0},A8={(p,q,w)∣q-p-2≥0,p>0,nnn(p+q)(q-p-2)wnnn-8p(q-p+1)>0},A9={(p,q,w)∣p-q-2≥0,q<0,nnn(p+q)(p-q-2)wnnn-8q(p-q+1)<0},A10={(p,q,w)∣q-p-2≥0,p<0,nnn(p+q)(q-p-2)wnnn-8p(q-p+1)<0},
and let
(54)S1={(p,q,w)∣p+q>0,2(p-q)2nnn-(2+w)(p+q)≥0,w≥0(p-q)2}∩(A1∪A2),S2={(p,q,w)∣p=q,w≥0}∪{(p,q,w)∣p≤2,q=0,max⁡⁡{0,2(p-1)}≤w}∪{(p,q,w)∣q≤2,p=0,max⁡⁡{0,2(q-1)}≤w}∪[{(p-q)2(p,q,w)∣p+q>0,w≥0,nnnn2(p-q)2-(2+w)(p+q)≤0}∩(A3∪A4)]∪[{(p-q)2(p,q,w)∣p+q<0,w≥0,nnnn2(p-q)2-(2+w)(p+q)>0}∩(A5∪A6)]∪[{(p-q)2(p,q,w)∣p+q>0,w≥0,nnnnn(p-q)2-3(p+q)+2≤0,nnnnn2(p-q)2-(2+w)(p+q)<0}nnnnn∩(A7∪A8)(p-q)2]∪[{(p,q,w)∣p+q<0,w≥0}∩(A9∪A10)].
The following assertions holds true.(1)
If(p,q,w)∈S1, then the generalized Muirhead-Heronian means Hp,q,w(x,y) is Schur-convex for (x,y)∈R++2.(2)
If(p,q,w)∈S2, then the generalized Muirhead-Heronian means Hp,q,w(x,y) is Schur-concave for (x,y)∈R++2.Proof.
Note that the expressionHp,q,w(x,y) is of symmetry between x and y and without loss of generality we assume that x≥y.
Case  1. If p=q, then Hp,q,w(x,y)=xy. Define
(55)(x-y)(∂H∂x-∂H∂y)=-(x-y)22xy≤0.
Hence,Hp,q,w(x,y) is Schur-concave for (x,y)∈R++2.
Case  2. If q=0, we have the following known results (see Theorem B and Remark 1 in Section 1).
H
p
,
0
,
w
(
x
,
y
) is Schur-convex for (x,y)∈R++2 if and only if
(56)(p,w)∈{(p,w)∣p≥2,0≤w≤2(p-1)}∪{(p,w)∣1≤p<2,w=0};
H
p
,
0
,
w
(
x
,
y
) is Schur-concave for (x,y)∈R++2 if and only if
(57)(p,w)∈{(p,w)∣p≤2,max⁡{0,2(p-1)}≤w}.Case  3. If q≠0, then
(58)(x-y)(∂H∂x-∂H∂y)=(x-y)Hp,q,w(x,y)F(x,y)xpyq+w(xy)(p+q)/2+xqyp,
where
(59)F(x,y)=yp+q-1p+q(p(xy)p-1+q(xy)q-1nnnnnnnnn+w(p+q)2(xy)((p+q)/2)-1nnnnnnnnn-q(xy)p-p(xy)qnnnnnnnnn-w(p+q)2(xy)(p+q)/2)=yp+q-1p+q(w(p+q)2pλp-1+qλq-1nnnnnnnnn+w(p+q)2λ((p+q)/2)-1-qλp-pλqnnnnnnnnn-w(p+q)2λ(p+q)/2)=λq-1yp+q-1p+q(w(p+q)2pλp-q+qnnnnnnnnn+w(p+q)2λ(p-q)/2-qλp-q+1nnnnnnnnn-pλ-w(p+q)2λ((p-q)/2)+1)=λq-1yp+q-1p+qg(λ),
where λ=x/y≥1; in addition, the definition of Hp,q,w(x,y) implies that p+q≠0.
Using Lemma6 gives
(60)(p+q)g(λ)≥0for(p,q,w)∈E11∪E12,
where
(61)E11=B1∩{2(p-q)2(p,q,w)∣q<0,nnnnnnnnn2(p-q)2-(2+w)(p+q)≥0},E12=B2∩{(p,q,w)∣q<0,nnnnnnnnnn2(p-q)2-(2+w)(p+q)≥0,nnnnnnnnnn(p+q)(p-q-2)wnnnnnnnnnn-8q(p-q+1)≥0}B1={(p,q,w)∣p+q>0,p-q-2≥0,w≥0},B2={(p,q,w)∣p+q>0,p-q-2<0,w≥0}.
On the other hand, we deduce from the symmetry ofHp,q,w(x,y) with respect to p and q that
(62)(p+q)g(λ)≥0for(p,q,w)∈E11′∪E12′,
where
(63)E11′=B1′∩{(p-q)2(p,q,w)∣p<0,nnnnnnnnn2(p-q)2-(2+w)(p+q)≥0},E12′=B2′∩{(p,q,w)∣p<0,nnnnnnnn2(p-q)2-(2+w)(p+q)≥0,nnnnnnnn(p+q)(q-p-2)w-8p(q-p+1)≥0},B1′={(p,q,w)∣p+q>0,nnnq-p-2≥0,w≥0},B2′={(p,q,w)∣p+q>0,nnnq-p-2<0,w≥0}.
Now, by using Lemma5 and combining the result stated in Case 2, we deduce that Hp,q,w(x,y) is Schur-convex under the conditions below:
(64)(p,q,w)∈E11∪E12∪E11′∪E12′∪{(p,q,w)∣q=0,p≥2,0≤w≤2(p-1)}∪{(p,q,w)∣q=0,1≤p<2,w=0}∪{(p,q,w)∣p=0,q≥2,0≤w≤2(q-1)}∪{(p,q,w)∣p=0,1≤q<2,w=0}={(p-q)2(p,q,w)∣p+q>0,nnnn2(p-q)2-(2+w)(p+q)≥0,w≥0}∩(A1∪A2)=S1.
This proves the validity of the first assertion in Theorem 9.
It is easy to find that(65)E21∪E22=A3∩{(p-q)2(p,q,w)∣p+q>0,w≥0,nnnnnnnnn2(p-q)2-(2+w)(p+q)≤0}.
In view of the symmetry ofHp,q,w(x,y) between p and q, utilizing a positional exchange between p and q in the above expression gives
(66)E21′∪E22′=A4∩{(p-q)2(p,q,w)∣p+q>0,w≥0,nnnnnnnn2(p-q)2-(2+w)(p+q)≤0}.
Hence, we deduce from Lemma7 that
(67)(p+q)g(λ)≤0forE21∪E22∪E21′∪E22′,
where
(68)E21∪E22∪E21′∪E22′={(p-q)2(p,q,w)∣p+q>0,w≥0,nnnnn2(p-q)2-(2+w)(p+q)≤0}∩(A3∪A4).
By using the same method as above, we can deduce that(69)(p+q)g(λ)≤0for(E23∪E24∪E23′∪E24′)∪(E31∪E31′)∪(E32∪E32′),
where
(70)E23∪E24∪E23′∪E24′={(p-q)2(p,q,w)∣p+q<0,w≥0,nnnnn2(p-q)2-(2+w)(p+q)>0}∩(A5∪A6),E31∪E31′={(p-q)2(p,q,w)∣p+q>0,w≥0,nnnnn(p-q)2-3(p+q)+2≤0,nnnnn2(p-q)2-(2+w)(p+q)<0}∩(A7∪A8),E32∪E32′={(p,q,w)∣p+q<0,w≥0}∩(A9∪A10).
Therefore, we deduce from Lemma5 and the results stated in Cases 1 and 2 that Hp,q,w(x,y) is Schur-concave under the following conditions:
(71)(p,q,w)∈(E21∪E22∪E21′∪E22′)∪(E23∪E24∪E23′∪E24′)∪(E31∪E31′)∪(E32∪E32′)∪{(p,q,w)∣p=q,w≥0}∪{(p,q,w)∣p≤2,q=0,max⁡⁡{0,2(p-1)}≤w}∪{(p,q,w)∣q≤2,p=0,max⁡⁡{0,2(q-1)}≤w}=S2.
This proves the validity of the second assertion in Theorem9. The proof of Theorem 9 is thus completed.


## 4. Some Remarks on the Results of Theorem9

In this section, we provide some remarks on the results given in Theorem9; we show that the sufficient conditions of Schur-convexity of the generalized Heronian means Hp,w(x,y) and the generalized Muirhead means M(p,q;x,y) can be deduced from Theorem 9 as special cases.Remark 10.
If we takeq=0 in Theorem 9, we have Hp,q,w(x,y)=Hp,w(x,y). Furthermore, we have
(72)S1={(p,q,w)∣p>0,q=0,0≤w≤2(p-1)}∩[{(p,q,w)∣p≥2,q=0,w≥0}nnn∪{(p,q,w)∣p<2,q=0,w=0}]={(p,q,w)∣p≥2,q=0,0≤w≤2(p-1)}∪{(p,q,w)∣1≤p<2,q=0,w=0},
which are the sufficient conditions of Schur-convex of the generalized Heronian means Hp,w(x,y) asserted by Theorem B.
On the other hand, we note that, forq=0,
(73){(p-q)2(p,q,w)∣p+q>0,w≥0,2(p-q)2-(2+w)(p+q)≤0}∩(A3∪A4)=∅,{(p-q)2(p,q,w)∣p+q<0,w≥0,nn2(p-q)2-(2+w)(p+q)>0}∩(A5∪A6)={(p,q,w)|p<0,q=0,w≥0}∩[{(p,q,w)∣p≤-2,q=0,(-p-2)w≤8(-p+1)}∪{(p,q,w)∣-2<p<0,q=0,w≥0}],{(p,q,w)∣p+q>0,(p-q)2-3(p+q)+2≤0,nn2(p-q)2-(2+w)(p+q)<0}∩(A7∪A8)=∅,{(p,q,w)∣p+q<0,w≥0}∩(A9∪A10)={(p,q,w)∣p<0,q=0,w≥0}∩{(p,q,w)∣p≤-2,q=0,nnnn(-p-2)w>8(-p+1)}.
Hence, the setS2 given in Theorem 9 reduces to the following form:
(74)S2={(p,q,w)∣p=0,q=0,w≥0}∪{(p,q,w)∣p≤2,q=0,max⁡⁡{0,2(p-1)}≤w}∪{(p,q,w)∣q=0,p=0,w≥0}∪{(p,q,w)∣p<0,q=0,w≥0}={(p,q,w)∣p≤2,q=0,max⁡{0,2(p-1)}≤w}.
These are the sufficient conditions of Schur-concave of the generalized Heronian meansHp,w(x,y) stated by Theorem B.Remark 11.
If we putw=0 in Theorem 9, we have Hp,q,w(x,y)=M(p,q;x,y). Furthermore, we have
(75)S1={(p,q,w)∣p+q>0,(p-q)2≥p+q,w=0}∩({(p,q,w)∣q≤0,w=0}nnn∪{(p,q,w)∣p≤0,w=0})={(p-q)2(p,q,w)∣p+q>0,pq≤0,nnnn(p-q)2≥p+q,w=0},
which are the sufficient conditions of Schur-convex of the generalized Muirhead means M(p,q;x,y) given by Theorem C.
In addition, forw=0, we have
(76){(p-q)2(p,q,w)∣p+q>0,w≥0,nn2(p-q)2-(2+w)(p+q)≤0}∩(A3∪A4)={(p-q)2(p,q,w)∣p>0,q>0,p≠q,nnnnn(p-q)2≤p+q,w=0},{(p-q)2(p,q,w)∣p+q<0,w≥0,nn2(p-q)2-(2+w)(p+q)>0}∩(A5∪A6)={(p,q,w)∣p+q<0,p≠q,w=0},{(p-q)2(p,q,w)∣p+q>0,(p-q)2-3(p+q)+2≤0,nn2(p-q)2-(2+w)(p+q)<0}∩(A7∪A8)=∅,{(p,q,w)∣p+q<0,w≥0}∩(A9∪A10)=∅.
Thus, the setS2 given in Theorem 9 reduces to the following form:
(77)S2={(p,q,w)∣p=q,w=0}∪{(p,q,w)∣p≤1,q=0,w=0}∪{(p,q,w)∣q≤1,p=0,w=0}∪{(p-q)2(p,q,w)∣p>0,q>0,p≠q,nnn(p-q)2≤p+q,w=0}∪{(p,q,w)∣p+q<0,p≠q,w=0}={(p,q,w)∣p≥0,q≥0,(p-q)2≤p+q,w=0}∪{(p,q,w)∣p+q<0,w=0}.
These are the sufficient conditions of Schur-concave of the generalized Muirhead meansM(p,q;x,y) asserted by Theorem C.



---

*Source: 706518-2014-08-27.xml*