# Schur Convexity and Inequalities for a Multivariate Symmetric Function

Authors: Ming-Bao Sun; Xin-Ping Li; Sheng-Fang Tang; Zai-Yun Zhang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In the article, we provide the Schur, Schur multiplicative, and Schur harmonic convexities properties for the symmetry functionFnx,r=Fnx1,x2,⋯,xn;r=∏1≤i1<i2<⋯<ir≤n∑j=1rxij/1−xij1/r on 0,1n and find several new analytical inequalities by use of the majorization theory, where x=x1,⋯,xn∈0,1n, r=1,2,⋯,n and i1,i2,⋯,in are positive integers.


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## Body

## 1. Introduction

Throughout the full article, 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
(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.Moreover, ifx=x1,x2,⋯,xn∈ℝ+n, then we denote
(2)logx=logx1,logx2,⋯,logxnxα=x1α,x2α,⋯,xnα.LetI⊆ℝ be a nonempty interval. Then a real-valued function f:I⟶ℝ is said to be convex (concave) on I if the inequality
(3)fλa+1−λb≤≥λfa+1−λfbholds for all a,b∈I and λ∈0,1.It is well known that the convex (concave) function is one of the most important functions in geometric function theory, and it has wide applications in mathematics and physics as well as in the fields of engineering technology [1–5]. Recently, the generalizations, extensions, and variants for the convexity (concavity) have attracted the attention of many researchers, for example, the the GA- and GG-convexities [6], s-convexity [7, 8], η-convexity [9], ρ-convexity [10], quasi-convexity [11], harmonic convexity [12, 13], exponential convexity [14], and generalized convexity [15]. In particular, many inequalities in pure and applied mathematics can be found in the literature [16–38] via the convexity (concavity) theory.As one of the variants of the convexity, the Schur convexity was introduced by Schur in 1923, and it becomes the subject of current research. In [39], Xia and Chu discussed the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity for the n-dimensional symmetry function
(4)Gnx,r=Gnx1,x2,⋯,xn;r=∏1≤i1<i2<⋯<ir≤n∑j=1rxij1+xij1r,on 0,1n and established several interesting inequalities by use the majorization theory, where i1,i2,⋯,in are positive integers.From the definition ofGnx,r given in (4), we clearly see that the key item λij=xij/1+xij∈0,1 in ∑j=1rxij/1+xij1/r for xij∈0,1. But in many practical problems, what we need is that the main term λij∈0,∞. If we replace λij=xij/1+xij with μij=xij/1−xij, then we clearly see that μij∈0,∞ for xij∈0,1.Motivated by the ideas given in [39] and the discussions above mentioned, we define the following symmetric function(5)Fnx,r=Fnx1,x2,⋯,xn;r=∏1≤i1<i2<⋯<ir≤n∑j=1rxij1−xij1r,discuss its Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity properties on 0,1n, and give their applications in the inequalities theory.


## 2. Definitions and Lemmas

To increase readability, we need recall some definitions in the beginning of this section. In order to establish our main results in the next section, we need introduce and establish several lemmas which we present in this section.Definition 1 (see [40]).
LetΩ⊆Rn be a nonempty set. Then, a real-valued function f defined on Ω is said to be Schur convex if
(6)fx1,x2,⋯,xn≤fy1,y2,⋯,yn,for each pair of n-tuples x=x1,x2,⋯,xn,y=y1,y2,⋯,yn∈Ω such that x<y, that is
(7)∑i=1kxi≤∑i=1kyik=1,2,⋯,n−1,∑i=1nxi=∑i=1nyi,where xi is the ith largest component of x. f is called Schur concave if −f is Schur convex.Definition 2 (see [40]).
LetΩ⊆R+n be a nonempty set. Then a real-valued function f defined on Ω is said to be Schur multiplicatively convex if
(8)fx1,x2,⋯,xn≤fy1,y2,⋯,yn,for each pair of n-tuples x=x1,x2,⋯,xn,y=y1,y2,⋯,yn∈Ω such that logx<logy. f is called Schur multiplicatively concave if 1/f is Schur multiplicatively convex.Definition 3 (see [40]).
LetΩ⊆R+n be a nonempty set. Then a real-valued function f defined on Ω is said to be Schur harmonic convex if
(9)fx1,x2,⋯,xn≤fy1,y2,⋯,yn,for each pair of n-tuples x=x1,x2,⋯,xn,y=y1,y2,⋯,yn∈Ω such that 1/x<1/y. f is called Schur harmonic concave if inequality (9) is reversed.Lemma 4 (see [40]).
LetΩ⊆Rn be a symmetric convex set with nonempty interior Ω∘ and f:Ω⟶R be a continuous symmetry function on Ω such that f is differentiable on Ω∘. Then f is Schur convex on Ω if and only if the inequality
(10)xi−xj∂fx∂xi−∂fx∂xj≥0holds for i,j=1,2,⋯,n and all x=x1,⋯,xn∈Ω∘. And f is Schur concave on Ω if and only if inequality (10) is reversed. Here, f being a symmetric function in Ω means that fPx=fx for any x∈Ω and any n×n permutation matrix P.Remark 5.
Sincef is symmetric, the Schur’s condition in Lemma 4, i.e., inequality (10) can be reduced to
(11)x1−x2∂fx∂x1−∂fx∂x2≥0.Lemma 6 (see [40]).
LetΩ⊆R+n be a symmetric multiplicatively convex set with nonempty interior Ω∘ and f:Ω⟶R+1 be a continuous symmetry function on Ω such that f is differentiable in Ω∘. Then f is Schur multiplicatively convex on Ω if and only if the inequality
(12)logx1−logx2x1∂fx∂x1−x2∂fx∂x2≥0is valid for all x=x1,⋯,xn∈Ω∘. And f is Schur multiplicatively concave on Ω if and only if inequality (12) is reversed. Here, Ω being a multiplicatively convex set means that x1/2y1/2∈Ω for x,y∈Ω.Lemma 7 (see [40]).
LetΩ⊆R+n be a symmetric harmonic convex set with nonempty interior Ω∘ and f:Ω⟶R+1 be a continuous symmetry function on Ω such that f is differentiable on Ω∘. Then f is Schur harmonic convex on Ω if and only if the inequality
(13)x1−x2x12∂fx∂x1−x22∂fx∂x2≥0takes place for all x=x1,⋯,xn∈Ω∘. And f is Schur harmonic concave in Ω if and only if inequality (13) is reversed. Here, Ω being a harmonic convex set means that 2xy/x+y∈Ω whenever x,y∈Ω.Lemma 8 (see [40]).
Letx=x1,x2,⋯,xn∈R+n, ∑i=1nxi=sn and c≥sn. Then,
(14)c−xnc/sn−1=c−x1nc/sn−1,c−x2nc/sn−1,⋯,c−xnnc/sn−1<x1,x2,⋯,xn=x.Lemma 9 (see [40]).
Letx=x1,x2,⋯,xn∈R+n, ∑i=1nxi=sn and c≥0. Then,
(15)c+xnc/sn+1=c+x1nc/sn+1,c+x2nc/sn+1,⋯,c+xnnc/sn+1<x1,x2,⋯,xn=x.Lemma 10 (see [40]).
Letx=x1,x2,⋯,xn∈Rn, ∑i=1nxi=sn and 0≤λ≤1. Then,
(16)sn−λxn−λ=sn−λx1n−λ,sn−λx2n−λ,⋯,sn−λxnn−λ<x1,x2,⋯,xn=x.Lemma 11.
Letfx1,x2,x3=3−2x1−2x2x3+x1+x2−1. Then, fx1,x2,x3≥0 for x=x1,x2,x3∈1/4,1/23∪3/4,13.Proof.
Leti=1,2,3. Then, we clearly see that ∂fnx,r/∂xi≤0 for x1,x2,x3∈3/4,13 and ∂fnx,r/∂xi≥0 for x1,x2,x3∈1/4,1/23, which lead to the conclusion that fx1,x2,x3 is decreasing on 3/4,1 and increasing on 1/4,1/2 with respect to each xi. Therefore, fx1,x2,x3≥f1,1,1=0 for x1,x2,x3∈3/4,13, and fx1,x2,x3≥f1/4,1/4,1/4=0 for x1,x2,x3∈1/4,1/23, that is fx1,x2,x3≥0 for x1,x2,x3∈1/4,1/23∪3/4,13.Similarly, we can easily derive Lemma12.Lemma 12.
Letr≥3 and fx1,x2=r/3−4/3x1+r/3−4/3x2+5/3−2/3r. Then, fx1,x2≤0 for x1,x2∈1/4,1/22.


## 3. Main Results

Theorem 13.
The symmetric functionFnx,r defined by (5) is Schur convex on 1/2r,1n if n≥2 and r∈1,2,⋯,n.Proof.
According to Lemma4 and Remark 5, we only need to prove that
(17)Pnx,r=x1−x2∂Fnx,r∂x1−∂Fnx,r∂x2≥0,for all x∈1/2r,1n, n≥2 and r∈1,2,⋯,n.We divide the proof into four cases.Case 1.
Ifr=1, then it follows from (5) that
(18)Fnx,1=∏i=1nxi1−xiand
(19)∂Fnx,1∂xi=Fnx,1xi1−xii=1,2.
From (19), we clearly see that
(20)Pnx,1=x1−x2∂Fnx,1∂x1−∂Fnx,1∂x2=x1−x22x1+x2−11−x11−x2Fnx,1≥0.Case 2.
Ifn≥2 and r=n, then from (5), we obtain
(21)Fnx,n=∑i=1nxi1−xi1n,and
(22)∂Fnx,n∂xi=1n1−xi2∑i=1nxi1−xi1n−1i=1,2.
Equation (22) leads to the conclusion that
(23)Pnx,n=x1−x2∂Fnx,n∂x1−∂Fnx,n∂x2=x1−x222−x1+x2n1−x121−x22∑i=1nxi1−xi1n−1≥0.Case 3.
Ifn≥3 and r=2, then from (5), we get
(24)Fnx,2=x11−x1+x21−x212∏j=3nx11−x1+xj1−xj12Fn−1x2,x3,⋯,xn;2=x11−x1+x21−x212∏j=3nx21−x2+xj1−xj12Fn−1x1,x3,⋯,xn;2,and
(25)∂Fnx,2∂xi=Fnx,221−xi21−x11−x2x1+x2−2x1x2+∑j=3n1−xi1−xjxi+xj−2xixj,for i=1,2.
Equation (25) leads to
(26)Pnx,2=x1−x2∂Fnx,2∂x1−∂Fnx,2∂x2=x1−x2221−x11−x2Fnx,2×2−x1+x2x1+x2−2x1x2+∑j=3nx1+x2−1+3−2x1−2x2xj1−xjx1+xj−2x1xjx2+xj−2x2xj.
It follows fromx∈1/4,1n that x1+x2−2x1x2=x11−x2+x21−x1≥0, x1+xj−2x1xj≥0, and x2+xj−2x2xj≥0. Let Δ=x1+x2−1+3−2x1−2x2xj. Then, it is easy to check that Δ≥0 for x=x1,x2,⋯,xn∈1/2,3/4n, and from Lemma 11, we know that Δ≥0 for x=x1,x2,⋯,xn∈1/4,1/2n∪3/4,1n. Therefore, Pnx,r≥0 for x∈1/4,1n follows from (26) immediately.Case 4.
Ifn≥4 and 3≤r≤n−1, then (5) leads to
(27)Fnx,r=∏3≤i3<⋯<ir≤nx11−x1+x21−x2+∑j=3rxij1−xij1r×∏3≤i2<⋯<ir≤nx11−x1+∑j=2rxij1−xij1rFn−1x2,⋯,xn;r=∏3≤i3<⋯<ir≤nx11−x1+x21−x2+∑j=3rxij1−xij1r×∏3≤i2<⋯<ir≤nx21−x2+∑j=2rxij1−xij1rFn−1x1,x3,⋯,xn;r,and
(28)∂Fnx,r∂xi=Fnx,rr1−xi2∑3≤i2<⋯<ir≤n1xi/1−xi+∑j=2rxij/1−xij+∑3≤i3<⋯<ir≤n1x1/1−x1+x2/1−x2+∑j=3rxij/1−xij,for i=1,2.
Equation (28) leads to the conclusion that
(29)Pnx,r=x1−x2∂Fnx,r∂x1−∂Fnx,r∂x2=−x1−x22r1−xi21−x22Fnx,r×∑3≤i2<⋯<ir≤n1−x1+x2+x1+x2−2∑j=2rxij/1−xijx1/1−x1+∑j=2rxij/1−xijx2/1−x2+∑j=2rxij/1−xij+∑3≤i3⋯<ir≤nx1+x2−2x1/1−x1+x2/1−x2+∑j=3rxij/1−xij.LetΘ=1−x1+x2+x1+x2−2∑j=2rxij/1−xij. Then, we only need to prove that Θ≤0 for x∈1/2r,1n. We divide the proof into three subcases.Subcase 1.
Ifx∈1/2,1n, then it is easy to check that Θ≤0.Subcase 2.
Ifx∈1/4,1/2n, then fx=x/1−x is increasing on 0,1, and from Lemma 12, we know that
(30)Θ≤1−x1+x2+13r−1x1+x2−2=53+r3−43x1+r3−43x2−23r≤0.Subcase 3.
Ifx∈1/2r,1/4n, then fx=x/1−x is increasing on 0,1 and we get
(31)Θ≤1−x1+x2+r−12r−1x1+x2−2≤1−2r−12r−1+x1+x2r−12r−1−1=12r−11−rx1+x2≤0.
From Subcases1–3, we know that Θ≤0 for all x∈1/2r,1n, and then from (29) we obtain Pnx,r≥0 for x∈1/2r,1n.
Cases1–4 lead to the conclusion that inequality (17) holds for all x∈1/2r,1n and r∈1,2,⋯,n. Therefore, Fnx,r is Schur convex on 1/2r,1n which follows from Lemma 4 and Remark 5.Remark 14.
From the proof of Cases1 and 2 in Theorem 13, we know that Fnx,1 is Schur concave on 0,1/2n and Fnx,n is Schur convex on 0,1/2nn.Theorem 15.
The symmetric functionFnx,r defined by (5) is Schur multiplicatively convex on 0,1n if n≥2 and r∈1,2,⋯,n.Proof.
According to Lemma6, we only need to prove that
(32)Qnx,r=logx1−logx2x1∂Fnx,r∂x1−x2∂Fnx,r∂x2≥0,for x∈0,1n and r∈1,2,⋯,n. We divide the proof into four cases.Case 1.
Ifr=1, then (19) leads to
(33)Qnx,1=logx1−logx2x1∂Fnx,1∂x1−x2∂Fnx,1∂x2=x1−x2logx1−logx21−x11−x2Fnx,1≥0.Case 2.
Ifn≥2 and r=n, then from (22), we get
(34)Qnx,n=logx1−logx2x1∂Fnx,n∂x1−x2∂Fnx,n∂x2=logx1−logx2x1−x21−x1x2n1−x121−x22∑i=1nxi1−xi1n−1≥0.Case 3.
Ifn≥3 and r=2, then from (25), we have
(35)Qnx,2=logx1−logx2x1∂Fnx,2∂x1−x2∂Fnx,2∂x2=logx1−logx2x1−x221−x11−x2Fnx,2×1−x1x2x1+x2−2x1x2+∑j=3n1−xjx1x2+xj−2x1x2xjx1+xj−2x1xjx2+xj−2x2xj.
Making use of the arithmetric-geometric mean inequality,(36)x1x2+xj−2x1x2xj≥2x1x2xj−x1x2xj,and the fact that ft=t−t>0 for t∈0,1, we get x1x2+xj−2x1x2xj≥0 for x∈0,1n.
Note thatx1+x2−2x1x2=x11−x2+x21−x1≥0, x1+xj−2x1xj≥0, and x2+xj−2x2xj≥0 for x∈0,1n. Therefore, Qnx,2≥0 follows from (35).Case 4.
Ifn≥4 and 3≤r≤n−1, then it follows from (28) that
(37)Qnx,r=logx1−logx2x1∂Fnx,r∂x1−x2∂Fnx,r∂x2=logx1−logx2x1−x2r1−xi21−x22Fnx,r×∑3≤i2<⋯<ir≤nx1x2+1−x1x2∑j=2rxij/1−xijx1/1−x1+∑j=2rxij/1−xijx2/1−x2+∑j=2rxij/1−xij+1−x1x2∑3≤i3<⋯<ir≤n1x1/1−x1+x2/1−x2+∑j=3rxij/1−xij≥0.
Therefore, inequality (32) follows from Cases 1–4, and Theorem 15 follows from Lemma 6 and (32).Theorem 16.
The symmetric functionFnx,r given in (5) is Schur harmonic convex on 0,1n if n≥2 and r∈1,2,⋯,n.Proof.
According to Lemma7, we only need to prove that
(38)Rnx,r=x1−x2x12∂Fnx,r∂x1−x22∂Fnx,r∂x2≥0,for x∈0,1n and r∈1,2,⋯,n. The proof is divided into four cases.Case 1.
Ifr=1, then from (19), we get
(39)Rnx,1=x1−x2x12∂Fnx,1∂x1−x22∂Fnx,1∂x2=x1−x221−x11−x2Fnx,1≥0.Case 2.
Ifn≥2 and r=n, then it follows from (22) that
(40)Rnx,n=x1−x2x12∂Fnx,n∂x1−x22∂Fnx,n∂x2=x1−x22x11−x2+x21−x1n1−x121−x22∑i=1nxi1−xi1n−1≥0.Case 3.
Ifn≥3 and r=2, then from (25), we obtain
(41)Rnx,2=x1−x2x12∂Fnx,2∂x1−x22∂Fnx,2∂x2=x1−x2221−x11−x2Fnx,2×1+∑j=3n1−xjx1x2+x1+x2xj−3x1x2xjx1+xj−2x1xjx2+xj−2x2xj.
Applying (41) and the arithmetric-geometric mean inequality
(42)x1x2+x1+x2xj−3x1x2xj≥3x12x22xj23−x1x2xj≥0,together with the facts that x1+xj−2x1xj=x11−xj+xj1−x1≥0 and x2+xj−2x2xj≥0 for x∈0,1n, we get Rnx,2≥0.Case 4.
Ifn≥4 and 3≤r≤n−1, then Equation (28) leads to
(43)Rnx,r=x1−x2x12∂Fnx,r∂x1−x22∂Fnx,r∂x2=x1−x22r1−xi21−x22Fnx,r×∑3≤i2<⋯<ir≤nx1x2+x1+x2−2x1x2∑j=2rxij/1−xijx1/1−x1+∑j=2rxij/1−xijx2/1−x2+∑j=2rxij/1−xij+∑3≤i3<⋯<ir≤nx1+x2−2x1x2x1/1−x1+x2/1−x2+∑j=3rxij/1−xij≥0.
From Cases1–4, we know that inequality (38) holds for x∈0,1n and r∈1,2,⋯,n.


## 4. Applications

In this section, we establish some new inequalities by the use of Theorems13, 15, and 16 and the majorization theory.Theorem 17.
Letn≥2, r∈1,2,⋯,n, x=x1,x2,⋯,xn, sn=∑i=1nxi, 0≤λ≤1, c1≥sn, and c2≥0. Then the following statements are true:
(1)
Ifx∈1/2r,1n, then(44)Fnx,r≥Fnc1−xnc1/sn−1,r,(45)Fnx,r≥Fnc2+xnc2/sn+1,r,(46)Fnx,r≥Fnsn−λxn−λ,r,and
(47)Fnx,r≥Fnsn+λxn+λ,r.(2)
Ifr=1 and x∈0,1/2n, then all the inequalities (44)–(47) are reversed; if r=n and x∈0,1/2nn, then all the inequalities (44)–(47) also holdProof.
Theorem17 follows from Theorems 13, Remark 14, and Lemmas 8–10 together with the fact that
(48)sn+λxn+λ=sn+λx1n+λ,sn+λx2n+λ,⋯,sn+λxnn+λ<x1,x2,⋯,xn=x.Theorem 18.
Letn≥2, r∈1,2,⋯,n, x=x1,x2,⋯,xn and Anx=1/n∑i=1nxi. Then, one has as follows:
(1)
Ifx∈1/2r,1n, then(49)∏1≤i1<i2<⋯<ir≤n∑j=1rxij1−xij1r≥rAnxAn1−xn!rn−r!r!(2)
Ifr=1 and x∈0,1/2n, then the inequality (49) is reversed; if r=n and x∈0,1/2nn, then the inequality (49) is also valid.Proof.
It is easy to see that(50)Anx,Anx,⋯,Anx<x1,x2,⋯,xn.
Therefore, Theorem18 follows from Theorem 13, Remark 14, and (50).Remark 19.
Letλ=0. Then Theorem 17 reduces to Theorem 18. On the other hand, if we take r=n, then Theorem 18 gives the well-known Shapiro inequality [41]:
(51)∑i=1nxi1−xi≥nAnx1−Anx,for x∈0,1n. If we take r=1, then Theorem 18(2) leads to the well-known Ky Fan inequality [42]:
(52)∏i=1nxi1−xi1n≤AnxAn1−x,for x∈0,1/2n.Theorem 20.
Ifn≥2, r∈1,2,⋯,n, x=x1,x2,⋯,xn∈0,1n, and Gnx=∏i=1nxi1/n, then
(53)∏1≤i1<i2<⋯<ir≤n∑j=1rxij1−xij1r≥rGnx1−Gnxn!rn−r!r!.Proof.
We clearly see that(54)logGnx,Gnx,⋯,Gnx<logx1,x2,⋯,xn.
Therefore, Theorem20 follows from Theorem 15 and (54).
Letr=1 and r=n. Then Theorem 20 leads to Corollary 21.Corollary 21.
Ifn≥2, x=x1,x2,⋯,xn∈0,1n, and Gnx=∏i=1nxi1/n, then one has
(55)∏i=1nxi1−xi≥Gnx1−Gnxn,and
(56)∑i=1nxi1−xi≥nGnx1−Gnx.Theorem 22.
Ifn≥2, r∈1,2,⋯,n, x=x1,x2,⋯,xn∈0,1n, and Hnx=n/∑i=1n1/xi, then we have
(57)∏1≤i1<i2<⋯<ir≤n∑j=1rxij1−xij1r≥rHnx1−Hnxn!rn−r!r!.Proof.
It is easy to see that(58)1Hnx,1Hnx,⋯,1Hnx<1x1,1x2,⋯,1xn.
Therefore, Theorem22 follows from Theorem 16 and (58) immediately.Theorem 23.
Suppose thatA∈MnCn≥2 is a complex matrix, and λ1,λ2,⋯,λn are the eigenvalues of A. If A is a positive definite Hermitian matrix, and I is the n×n unit matrix, then the inequalities
(59)∏1≤i1<i2<⋯<ir≤n∑j=1rλijtrA−λij1r≥rdetAntrA−detAnn!rn−r!r!,(60)∏1≤i1<i2<⋯<ir≤n∑j=1rλij1r≥rdetAndetI+An−detAnn!rn−r!r!,and
(61)∏1≤i1<i2<⋯<ir≤n∑j=1r1λij1r≥nrtrAn!rn−r!r!
hold for allr∈1,2,⋯,n.Proof.
It is not difficult to verify that(62)logdetAntrA,detAntrA,⋯,detAntrA<logλ1trA,λ2trA,⋯,λntrA,(63)logdetAdetI+An,detAdetI+An,⋯,detAdetI+An<logλ11+λ1,λ21+λ2,⋯,λn1+λn,and
(64)trA+In,trA+In,⋯,trA+In<1+λ1,1+λ2,⋯,1+λn.
Therefore, inequalities (59) and (60) follow from Theorems 15, (62), and (63), and inequality (61) follows from Theorem 16 and (64).Theorem 24.
LetA=A1A2⋯An+1 be an n-dimensional simplex of Rn, P be an arbitrary point in the interior of A, and Bi be the intersection point of straight line AiP and hyperplane ∑i=A1A2⋯Ai−1Ai+1⋯An+1i=1,2,⋯,n+1. Then, the inequalities
(65)∏1≤i1<i2<⋯ir≤n+1∑j=1r2rAijBij2r−1PBij−11r≥r2nr+12r−1n+1!rn−r+1!r!,and
(66)∏1≤i1<i2<⋯ir≤n+1∑j=1r2rAijBij2r−1PAij−11r≥rn+2r2nr−nn+1!rn−r+1!r!
hold for allr∈1,2,⋯,n+1.Proof.
It is easy to see that∑i=1n+1PBi/AiBi=1 and PAi/AiBi=1−PBi/AiBi for i=1,2,⋯,n+1, which imply that
(67)2nr+12rn+1,2nr+12rn+1,⋯,2nr+12rn+1<1−2r−1PB12rA1B1,1−2r−1PB22rA2B2,⋯,1−2r−1PBn+12rAn+1Bn+1,(68)1−2r−1PBi2rAiBi∈12r,1,(69)n+2r2rn+1,n+2r2rn+1,⋯,n+2r2rn+1<1−2r−1PA12rA1B1,1−2r−1PA22rA2B2,⋯,1−2r−1PAn+12rAn+1Bn+1,and
(70)1−2r−1PAi2rAiBi∈12r,1.
Therefore, inequality (65) follows from (67), (68), and Theorem 13, and inequality (66) follows from (69), (70), and Theorem 13.Remark 25.
Mitrinović et al. [41] established a series of inequalities forPAi/AiBi and PBi/AiBii=1,2,⋯,n,n+1. Obviously, our inequalities in Theorem 24 are different from theirs.


## 5. Conclusions

We have presented the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity properties for the symmetric functionFnx,r defined by (5). As applications, we have found several new analytical inequalities by the use of the majorization theory. In particular, we have provided a common generalization for the Sharpiro and the Ky Fan inequalities by the use of our obtained results. Our results are the generalizations and improvements of the previously known results, and our ideas and approach may lead to a lot of follow-up research.



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*Source: 9676231-2020-07-10.xml*