# The Schur Multiplicative and Harmonic Convexities for Three Classes of Symmetric Functions

Authors: Ming-bao Sun; Xin-ping Li; Ying-hui Zhang; Zai-yuan Zhang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We investigate the Schur harmonic convexity for two classes of symmetric functions and the Schur multiplicative convexity for a class of symmetric functions by using a new method and generalizing previous result. As applications, we establish some inequalities by use of the theory of majorization, in particular, and we give some new geometric inequalities in then-dimensional space.


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## Body

## 1. Introduction

Throughout this paper, we denote byRn the n-dimensional Euclidean space, R=R1=(-∞,+∞) and R+n=x=x1,x2,…,xn∈Rn:xi>0,i=1,2,…,n. For x=(x1,x2,…,xn),y=(y1,y2,…,yn)∈Rn, and α∈R, we denote by (1)x+y=x1+y1,x2+y2,…,xn+yn,xy=x1y1,x2y2,…,xnyn,αx=αx1,αx2,…,αxn,α+x=α+x1,α+x2,…,α+xn,1x=1x1,1x2,…,1xn.Furthermore, for x=(x1,x2,…,xn)∈R+n, we denote by (2)log⁡x=log⁡x1,log⁡x2,…,log⁡xn,xα=x1α,x2α,…,xnα.Recently, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity for some special functions and their applications have been investigated by many authors; see, for instance, [1–12] and the references therein.Forx=(x1,…,xn)∈[0,1)n=x=x1,x2,…,xn∈Rn:0≤xi<1,i=1,2,…,n and r∈1,2,…,n, the symmetric function Fn(x,r) was defined by Guan [13] as (3)Fnx,r=Fnx1,x2,…,xn;r=∑1≤i1<i2<⋯ir≤n∏j=1rxij1-xij,where i1,i2,…,in are positive integers. In [13], Guan discussed the Schur convexity and Schur multiplicative convexity for Fn(x,r), established some interesting inequalities by use of the theory of majorization, and proposed an open problem as follows.Open Problem A. If n≥5 and r∈3,…,n-2, then the function Fn(x,r) is Schur convex, or Schur concave.Chu et al. [2] solved the Open Problem A and established the following Theorem A.Theorem A.
Ifn≥2 and r∈2,…,n, then the function Fn(x,r) is Schur concave in 0,r-1/2(n-1)n and Schur convex in r-1/2(n-1),1n.Xia and Chu [14] defined the following symmetric function: (4)Gnx,r=Gnx1,x2,…,xn;r=∑1≤i1<i2<⋯ir≤n∏j=1r1-xijxij,where x=(x1,…,xn)∈(0,1]n, r∈1,2,…,n, and i1,i2,…,in are positive integers. In [14], they discussed the Schur convexity and Schur multiplicative convexity for Gn(x,r) and established some inequalities.Forx=(x1,…,xn)∈R+n and r∈1,2,…,n, the symmetric function Mn(x,r) was defined by Xia et al. [15] as (5)Mnx,r=Mnx1,x2,…,xn;r=∑1≤i1<i2<⋯ir≤n∏j=1rxij1+xij,where i1,i2,…,in are positive integers. In [15], Xia et al. proved that Mn(x,r) is Schur concave and Schur harmonic convex in R+n (see [15, Theorems 3.1-3.2]) and Schur multiplicatively concave in [1,+∞)n, while Mn(x,1) is Schur multiplicatively convex in (0,1]n, and Mn(x,n) is Schur multiplicatively concave in [1,+∞)n (see [15, Theorem 3.3]).It is natural to ask whether the symmetric functionsFn(x,r) and Gn(x,r) are Schur harmonic convex, or Schur harmonic concave in R+n, while the function Mn(x,r) is Schur multiplicatively convex or Schur multiplicatively concave for x∈(0,1)n and r∈2,…,n-1, which have not been studied in existing literatures.The main purpose of this paper is to discuss the Schur harmonic convexity ofFn(x,r) and Gn(x,r) in R+n and solve the problem of the Schur multiplicative convexity or concavity of Mn(x,r) in (0,1)n(r=2,…,n-1) by using a new method, the main difficulty in proving Theorem 3 is to apply some techniques of differential calculus and combinatorics, in particular, and construct one complicate equality (29) in proof of Theorem 3. As applications, we establish some inequalities by use of the theory of majorization, in particular, we give some new geometric inequalities in the n-dimensional space.Our main results are the following three theorems.Theorem 1.
Forx∈(0,1)n∪(1,+∞)n, n≥2 and r∈1,2,…,n.
(1) The functionFn(x,r) is Schur harmonic convex in (0,1)n;
(2) Ifr is even integer (or odd integer, respectively), then Fn(x,r) is Schur harmonic convex (or concave, respectively) in (1,+∞)n.Theorem 2.
Forx∈(0,+∞)n, n≥2 and r∈1,2,…,n.
(1) The functionGn(x,r) is Schur harmonic concave in (0,1]n;
(2) ifr is even integer (or odd integer, respectively), then Gn(x,r) is Schur harmonic concave (or convex, respectively) in (1,+∞)n.Theorem 3.
Forx∈(0,+∞)n, n≥3, and r∈2,…,n-1, Mn(x,r) is Schur multiplicatively convex in (0,n-r/n-1]n and Schur multiplicatively concave in (n-r/n-1,+∞)n.Remark 4.
From the proof of Theorem 3.3 in [15], it is easy to see that Theorem 3 is valid for r=1 and r=n, and Theorems 3 is a generalization of the Theorem 3.3 in [15], which is the main result in [15], thus, by using a new method we generalize and solve the problem of the Schur multiplicative convexity or concavity of Mn(x,r) in (0,1)n(r=2,…,n-1) from [15].This paper, except for the introduction, is divided into three sections. In Section2, we recall some definitions and lemmas. By using the results of Section 2, we give the proof of the main results in Section 3. Finally, some applications are given by use of the theory of majorization.


## 2. Definitions and Lemmas

For convenience, we recall some definitions.Definition 5.
LetΩ⊆Rn be a set; a real-valued function f on Ω is said to be Schur convex if (6)fx1,x2,…,xn≤fy1,y2,…,ynfor each pair of n-tuples x=(x1,x2,…,xn) and y=(y1,y2,…,yn) in Ω, such that x≺y, that is (7)∑i=1kxi≤∑i=1kyi,k=1,2,…,n-1and (8)∑i=1nxi=∑i=1nyi,where x[i] denotes the ith largest component in x. f is called Schur concave if -f is Schur convex.The above notion of Schur convexity was first introduced by Schur in 1923 and has many important applications in analytic inequalities [16–20], isoperimetric problem for polytopes [21], linear regression [22], combinatorial optimization [23], graphs and matrices [24], gamma and digamma functions [25], information-theoretic topics [26], stochastic orderings [27], and other related fields. The Definition 5 can be found in many references such as [20, 28]. Following Schur, the following definition of the Schur multiplicative convex function was introduced in [29, 30].Definition 6.
LetΩ⊆R+n be a set; a real-valued function f on Ω is said to be Schur multiplicatively convex if (9)fx1,x2,…,xn≤fy1,y2,…,ynfor each pair of n-tuples x=(x1,x2,…,xn) and y=(y1,y2,…,yn) in Ω, such that log⁡x≺log⁡y. f is called Schur multiplicatively concave if 1/f is Schur multiplicatively convex.Recently, Chu and Lv [31] introduced the notion of Schur harmonic convexity.Definition 7.
LetΩ⊆R+n be a set; a real-valued function f on Ω is said to be Schur harmonic convex function if (10)fx1,x2,…,xn≤fy1,y2,…,yn,for each pair of n-tuples x=(x1,x2,…,xn) and y=(y1,y2,…,yn) in Ω, such that 1/x≺1/y. f is called Schur harmonic concave if inequality (10) is reversed.In order to establish our main result, we need two lemmas.Lemma 8 (see [29, 30]).
LetΩ⊆R+n be a symmetric multiplicatively convex set with nonempty interior in int⁡Ω and f:Ω→R+1 be a continuous symmetry function on Ω. If f is differentiable in int⁡Ω, then f is Schur multiplicatively convex on Ω if and only if (11)log⁡x1-log⁡x2x1∂fx∂x1-x2∂fx∂x2≥0for all x=(x1,…,xn)∈int⁡Ω. And f is Schur multiplicatively concave on Ω if and only if inequality (11) is reversed. Here Ω is a multiplicatively convex set which means that x1/2y1/2∈Ω for x,y∈Ω.Lemma 9 (see [31]).
LetΩ⊆R+n be a symmetric harmonic convex set with nonempty interior in int⁡Ω and f:Ω→R+1 be a continuous symmetry function on Ω. If f is differentiable on int⁡Ω, then f is Schur harmonic convex on Ω if and only if (12)x1-x2x12∂fx∂x1-x22∂fx∂x2≥0for all x=(x1,…,xn)∈int⁡Ω. And f is Schur harmonic concave in Ω if and only if inequality (12) is reversed. Here Ω is a harmonic convex set which means that 2xy/x+y∈Ω for x,y∈Ω.


## 3. Proofs of the Main Results

Proof of Theorem1.
(1) According to Lemma9, we only need to prove that (13)Pnx,r=x1-x2x12∂Fnx,r∂x1-x22∂Fnx,r∂x2≥0for all x∈(0,1)n and r∈1,2,…,n. The proof is divided into three cases.
Case 1. If r=1, then from (3) we have (14)Fnx,1=∑i=1nxi1-xi,(15)∂Fnx,1∂xi=11-xi2.i=1,2By (15), we obtain (16)Pnx,1=x1-x2x12∂Fnx,1∂x1-x22∂Fnx,1∂x2=x1-x22x11-x2+x21-x11-x121-x22≥0.
Case 2. If n≥2 and r=n, then from (3) we obtain (17)Fnx,n=∏i=1nxi1-xi,(18)∂Fnx,n∂xi=Fnx,nxi1-xi.i=1,2By (18), we obtain (19)Pnx,n=x1-x2x12∂Fnx,n∂x1-x22∂Fnx,n∂x2=x1-x22Fnx,n1-x11-x2≥0.
Case 3. If n≥3 and 2≤r≤n-1, noting that the r-th order elementary symmetric function En(x,r) (see [32]) is defined as (20)Enx,r=Enx1,x2,…,xn;r=∑1≤i1<i2<⋯<ir≤n∏j=1rxij,where x=(x1,…,xn)∈R+n, r∈1,2,…,n, and i1,i2,…,in are positive integers and En(x,0)=En(x1,x2,…,xn;0)=1.
Letui=xi/1-xi,i=1,…,n, and then from (3), we have(21)Fnx,r=x1x21-x11-x2En-2u3,u4,…,un;r-2+x11-x1En-2u3,u4,…,un;r-1+En-1u2,…,un;r=x1x21-x11-x2En-2u3,u4,…,un;r-2+x21-x2En-2u3,u4,…,un;r-1+En-1u1,u3,…,un;r,(22)∂Fnx,r∂x1=x21-x121-x2En-2u3,u4,…,un;r-2+11-x12En-2u3,u4,…,un;r-1,(23)∂Fnx,r∂x2=x11-x11-x22En-2u3,u4,…,un;r-2+11-x22En-2u3,u4,…,un;r-1,From (22) and (23), we obtain (24)Pnx,r=x1-x2x12∂Fnx,r∂x1-x22∂Fnx,r∂x2=x1-x221-x121-x2)2x1x2En-2u3,u4,…,un;r-2+x11-x2+x21-x1En-2u3,u4,…,un;r-1≥0,
From Cases 1–3 we obtain that (13) holds for all x∈(0,1)n and r∈1,2,…,n; therefore, Fn(x,r) is schur harmonic convex in (0,1)n.
(2) Here we give only the proof in the case ofr being even integer, since the proof in the case of r being odd integer is similar. According to Lemma 9, we only need to prove that (13) holds for all x∈(1,+∞)n. From the proofs of Cases 2-3 in (1), it is easy to see that Pn(x,r)≥0 for x∈(1,+∞)n; thus we derive that (13) holds for all x∈(1,+∞)n in the case of r being even integer; therefore, Fn(x,r) is Schur harmonic convex in (1,+∞)n.
The proof of Theorem1 is completed.Similar to the proof of Theorem1, we can use Lemma 9 and (20) to prove Theorem 2; therefore we omit the details of the proof. Next we give the proof of Theorem 3.Proof of Theorem3.
According to Lemma8, we only need to discuss the nonnegativity and nonpositivity of Qn(x,r)=(log⁡x1-log⁡x2)x1∂Mn(x,r)/∂x1-x2∂Mn(x,r)/∂x2 for all x∈(0,+∞)n and r∈2,…,n-1.
Letvi=xi/1+xi,i=1,…,n; then from (5) and (20) we have (25)Mnx,r=x1x21+x11+x2En-2v3,v4,…,vn;r-2+x11+x1En-2v3,v4,…,vn;r-1+En-1v2,…,vn;r=x1x21+x11+x2En-2v3,v4,…,vn;r-2+x21+x2En-2v3,v4,…,vn;r-1+En-1v1,v3,…,vn;r,(26)∂Enx,r∂x1=1+x2x1fx1fx2En-2v3,v4,…,vn;r-2+1x1fx1En-2v3,v4,…,vn;r-1,(27)∂Enx,r∂x2=1+x1x2fx1fx2En-2v3,v4,…,vn;r-2+1x2fx2En-2v3,v4,…,vn;r-1,where f(x)=(1+x)2/x. From (26) and (27) we obtain (28)Qnx,r=log⁡x1-log⁡x2x1∂Mnx,r∂x1-x2∂Mnx,r∂x2=-log⁡x1-log⁡x2fx1fx2gx1,r-gx2,r,where g(x,r)=xEn-2(v3,v4,…,vn;r-2)+f(x)En-2(v3,v4,…,vn;r-1). Note that (29)En-2v3,v4,…,vn;r-1=1r-1∑3≤i3<⋯<ir≤n∏j=3rxij1+xij∑ik≠ijj=3,…,nik∈3,…,nxik1+xik,(30)f′x=x2-1x2,(31)f′′x=2x-3,(32)∂gx,r∂x=En-2v3,v4,…,vn;r-2+f′xEn-2v3,v4,…,vn;r-1=∑3≤i3<⋯<ir≤n∏j=3rxij1+xij1+1r-1f′x∑ik≠ijj=3,…,nik∈3,…,nxik1+xik,and (33)∂2gx,r∂x2=f′′xEn-2v3,v4,…,vn;r-1.
LetA=n-r/n-1; then 0<A<1, from (30) we have (34)S=1+n-rr-1f′AA1+A=0.From (31), we have f′′(x)>0 for x∈(0,+∞); hence by (33), if xi∈(0,+∞)(i=3,4,…,n), then ∂2g(x,r)/∂x2>0 for x∈(0,+∞); therefore, ∂g(x,r)/∂x is increasing on (0,+∞) with x; further from (32) and (34), we obtain for x,xi∈0,A(i=3,4,…,n)(35)∂gx,r∂x<∂gA,r∂x=∑3≤i3<⋯<ir≤n∏j=3rxij1+xij1+1r-1f′A∑ik≠ijj=3,…,nik∈3,…,nxik1+xik≤SMn-2A,A,…,A;r-2=0and have for x,xi∈(A,+∞) (i=3,4,…,n) (36)∂gx,r∂x>∂gA,r∂x=∑3≤i3<⋯<ir≤n∏j=3rxij1+xij1+1r-1f′A∑ik≠ijj=3,…,nik∈3,…,nxik1+xik≥SMn-2A,A,…,A;r-2=0.Hence by (35) and (36), respectively, if xi∈(0,A)(i=3,4,…,n), then g(x,r) is decreasing on (0,A) with x, and if xi∈(A,+∞)(i=3,4,…,n), then g(x,r) is increasing on (A,+∞) with x; further from (28) we derive that Qn(x,r)≥0 for x∈(0,A)n, and Qn(x,r)≤0 for x∈(A,+∞)n. Therefore, from Lemma 8, Mn(x,r) is Schur multiplicatively convex in (0,n-r/n-1]n and Schur multiplicatively concave in (n-r/n-1,+∞)n.
The proof of Theorem3 is completed.Remark 10.
It is easy to see that the key point of our proof of Theorem3 is to construct one monotone function g(x,r) and prove that g(x,r) is decreasing on (0,A), and increasing on (A,+∞) with x, which is different from the proof of Theorem 3.3 in [15].


## 4. Applications

In this section, we establish some inequalities by use of Theorems1–3 and the theory of majorization.Theorem 11.
Letn≥2, x=(x1,x2,…,xn), An(x)=1/n∑i=1nxi, and r∈1,2,…,n.
(1) Suppose thatx∈(0,1)n. If r is even integer, then (37)∑1≤i1<i2<⋯<ir≤n∏j=1r1xij-1≥n!r!n-r!1Anx-1r,(38)∑1≤i1<i2<⋯<ir≤n∏j=1rxij-1≤n!r!n-r!Anx-1r,while if r is odd integer, then the inequalities (37) and (38) are reversed
(2) Ifx∈(1,+∞)n, the inequalities (37) and (38) hold.Proof.
It is easy to see that(39)Anx,Anx,…,Anx≺x1,x2,…,xn.
From Theorems1 and 2, respectively, and applying (39), we get that Theorem 11 holds.Theorem 12.
Letn≥2, x=(x1,x2,…,xn), Hn(x)=n/∑i=1n1/xi, and r∈1,2,…,n.
(1) Ifx∈(0,1)n, then (40)∑1≤i1<i2<⋯<ir≤n∏j=1rxij1-xij≥n!n-r!r!HnxHnx-1r,(41)∑1≤i1<i2<⋯<ir≤n∏j=1r1-xijxij≤n!n-r!r!1-HnxHnxr;(2) Suppose that x∈(1,+∞)n. If r is even integer, then the inequalities (40) and (41) hold, while if r is odd integer, the inequalities (40) and (41) are reversed.Proof.
It is easy to see that(42)1Hnx,1Hnx,…,1Hnx≺1x1,1x2,…,1xn.From Theorems 1 and 2, respectively, and applying (42), we obtain that Theorem 12 holds.Theorem 13.
Letn≥2, x=(x1,x2,…,xn), Gn(x)=∏i=1nxi1/n, and r∈1,2,…,n. If x∈(0,n-r/n-1]n, then (43)∑1≤i1<i2<⋯ir≤n∏j=1rxij1+xij≥n!n-r!r!Gnx1+Gnxr,if x∈[n-r/n-1,+∞)n, the inequality (43) is reversed.Proof.
It is easy to see that(44)log⁡Gnx,Gnx,…,Gnx≺log⁡x1,x2,…,xn.From Theorems 3 and Remark 4, and applying (44), we obtain that Theorem 13 holds.Remark 14.
It is easy to see that Theorem13 is a generalization of the Theorem 4.5 in [15].Theorem 15.
LetA=A1A2⋯An+1 be an n-dimensional simples in Rn and P be an arbitrary point in the interior of A. If Bi is the intersection point of straight AiP and hyperplane ∑i=A1A2⋯Ai-1Ai+1⋯An+1(i=1,2,…,n+1). Then for r∈1,2,…,n+1(45)∑1≤i1<i2<⋯<ir≤n+1∏j=1rAijBijPAij≥n+1!n+1-r!r!n+1nr,(46)∑1≤i1<i2<⋯<ir≤n+1∏j=1rPAijAijBij≤n+1!n+1-r!r!nn+1r,(47)∑1≤i1<i2<⋯<ir≤n+1∏j=1rAijBijPBij≥n+1!n+1-r!r!n+1r,(48)∑1≤i1<i2<⋯<ir≤n+1∏j=1rPBijAijBij≤n+1!n+1-r!r!1n+1r,Proof.
It is easy to see that∑i=1n+1PBi/AiBi=1 and PAi/AiBi=1-PBi/AiBi,i=1,2,…,n+1; these imply that (49)1n+1,1n+1,…,1n+1≺PB1A1B1,PB2A2B2,…,PBn+1An+1Bn+1,(50)nn+1,nn+1,…,nn+1≺PA1A1B1,PA2A2B2,…,PAn+1An+1Bn+1.
Therefore, (45)-(48) follow from Theorems 1(2) and 2(2), respectively, together with (49) and (50).Remark 16.
The inequalities (45)-(48) can be found in [33, p.473-480]; here we give a different proof of them.Theorem 17.
LetA=A1A2⋯An+1 be an n-dimensional simples in Rn and r be the inradius of A. Let hi be the altitude of A from the vertex Ai, i.e., the distance from Ai to hyperplane ∑i=A1A2⋯Ai-1Ai+1⋯An+1(i=1,2,…,n+1), and ri be the radius of i-th escribed hypersphere of A. Then for k∈1,2,…,n+1(51)∑1≤i1<i2<⋯<ik≤n+1∏j=1krijrij-r≥n+1!n+1-k!k!n+12k,(52)∑1≤i1<i2<⋯<ik≤n+1∏j=1krij-rrij≤n+1!n+1-k!k!2n+1k,(53)∑1≤i1<i2<⋯<ik≤n+1∏j=1khijhij-r≥n+1!n+1-k!k!n+1nk,(54)∑1≤i1<i2<⋯<ik≤n+1∏j=1khij-rhij≥n+1!n+1-k!k!nn+1k,Proof.
LetV denotes the volume of A, Fi(i=1,2,…,n+1) denote the (n-1)-dimensional volume of the side face of A opposite to the vertex Ai (i=1,2,…,n+1). Let F=∑i=1n+1Fi; noting the well-known fact (see [33, p.463]) (55)r=nVF,hi=nVFi,ri=nVF-2Fi,we obtain (56)rri=F-2FiF,rhi=FiF.Thus we have (57)∑i=1n+1rri=n-1,∑i=1n+1rhi=1,and further we obtain (58)n-1n+1,n-1n+1,…,n-1n+1≺rr1,rr2,…,rrn+1,(59)1n+1,1n+1,…,1n+1≺rh1,rh2,…,rhn+1.
Therefore, (51)-(54) follow from Theorems 1(2) and 2(2), respectively, together with (58) and (59).Remark 18.
The inequality (51) can be found in the Proposition 4 from [19]; here we give a different proof of it. Mitrinović, Pečarić, and Volenec (see [33, p.463-473]) established a series of inequalities for r, hi, and ri; obviously, our inequalities (52)-(54) in Theorem 17 are different from theirs.


## 5. Conclusions

In this paper, we investigate the Schur harmonic convexity for two classes of symmetric functions defined by Guan (2007), Xia and Chu (2009), and the Schur multiplicative convexity for a class of symmetric functions defined by Xia et al. (2010) by using a new method and generalize the main results of Xia et al. (2010). As applications, we establish some inequalities by use of the theory of majorization; in particular, we give some new geometric inequalities in then-dimensional space. It is easy to see that(1) Our Theorem13 is a generalization of the Theorem 4.5 in Xia et al. (2010).(2) The inequalities (45)-(48) can be found in Mitrinović (1989); here we give a different proof of them.(3) The inequality (51) can be found in the Proposition 4 from Wu (2005); here we give a different proof of it.



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