# New proofs of Schur-concavity for a class of symmetric functions

## Abstract

By properties of the Schur-convex function, Schur-concavity for a class of symmetric functions is simply proved uniform.

2000 Mathematics Subject Classification: Primary 26D15; 05E05; 26B25.

## 1. Introduction

Throughout the article, denotes the set of real numbers, x= (x1, x2, ..., x n ) denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as

$\begin{array}{c}{ℝ}^{n}=\left\{x=\left({x}_{1},...,{x}_{n}\right):{x}_{i}\in ℝ,i=1,...,n\right\},\\ {ℝ}_{+}^{n}=\left\{x=\left({x}_{1},...,{x}_{n}\right):{x}_{i}>0,i=1,...,n\right\}.\end{array}$

In particular, the notations and + denote 1 and ${ℝ}_{+}^{1}$ respectively.

For convenience, we introduce some definitions as follows.

Definition 1. [1, 2] Let x =(x1, ..., x n ) and y =(y1, ..., y n ) n.

(i) xy means x i y i for all i = 1, 2,..., n.

(ii) Let Ω n, φ: Ω → is said to be increasing if xy implies φ(x) ≥ φ(y ). φ is said to be decreasing if and only if is increasing.

Definition 2. [1, 2] Let x =(x1, ..., x n ) and y = (y1, ..., y n ) n.

(i) x is said to be majorized by y(in symbols x y) if ${\sum }_{i=1}^{k}{x}_{\left[i\right]}\le {\sum }_{i=1}^{k}{y}_{\left[i\right]}$ for k = 1, 2,..., n - 1 and ${\sum }_{i=1}^{n}{x}_{i}={\sum }_{i=1}^{n}{y}_{i}$, where x[1] ≥ · · · ≥ x[n]and y[1] ≥ · · · ≥ y[n]are rearrangements of x and y in a descending order.

(ii) Let Ω n, φ: Ω → is said to be a Schur-convex function on Ω if x y on Ω implies φ (x) ≤ φ (y). φ is said to be a Schur-concave function on Ω if and only if is Schur-convex function on Ω.

Definition 3. [1, 2] Let x = (x1, ..., x n ) and y= (y1, ..., y n ) n.

(i) Ω nis said to be a convex set if x, y Ω, 0 ≤ α ≤ 1 implies α x+ (1- α)y =(αx1 + (1 - α)y1, ...,αx n + (1- α)y n ) Ω.

(ii) Let Ω nbe convex set. A function φ: Ω → is said to be a convex function on Ω if

$\phi \left(\alpha x+\left(1-\alpha \right)y\right)\le \alpha \phi \left(x\right)+\left(1-\alpha \right)\phi \left(y\right)$

for all x, y Ω, and all α [0,1]. φ is said to be a concave function on Ω if and only if is convex function on Ω.

Recall that the following so-called Schur's condition is very useful for determining whether or not a given function is Schur-convex or Schur-concave.

Theorem A. [[1], p. 5] Let Ω nis symmetric and has a nonempty interior convex set. Ω0is the interior of Ω. φ: Ω → is continuous on Ω and differentiable in Ω0. Then φ is the Schur-convex (Schur-concave) function, if and only if φ is symmetric on Ω and

$\left({x}_{1}-{x}_{2}\right)\left(\frac{\partial \phi }{\partial {x}_{1}}-\frac{\partial \phi }{\partial {x}_{2}}\right)\ge 0\left(\le 0\right)$
(1)

holds for any x Ω0.

In recent years, by using Theorem A, many researchers have studied the Schur-convexity of some of symmetric functions.

Chu et al. [3] defined the following symmetric functions

${F}_{n}\left(x,k\right)=\prod _{1\le {i}_{1}<...<{i}_{k}\le n}\frac{{\sum }_{j=1}^{k}{x}_{{i}_{j}}}{{\sum }_{j=1}^{k}\left(1+{x}_{{i}_{j}}\right)},k=1,...,n,$
(2)

and established the following results by using Theorem A.

Theorem B. For k = 1,..., n, F n (x , k) is an Schur-concave function on${ℝ}_{+}^{n}$.

Jiang [4] are discussed the following symmetric functions

${H}_{k}^{*}\left(x\right)=\prod _{1\le {i}_{1}<...<{i}_{k}\le n}{\sum }_{j=1}^{k}{x}_{{i}_{j}}^{1/k},k=1,...,n,$
(3)

and established the following results by using Theorem A.

Theorem C. For$k=1,...,n,\phantom{\rule{2.77695pt}{0ex}}{H}_{k}^{*}\left(x\right)$is an Schur-concave function on${ℝ}_{+}^{n}$.

Xia and Chu [5] investigated the following symmetric functions

${\varphi }_{n}\left(x,k\right)=\prod _{1\le {i}_{1}<...<{i}_{k}\le n}{\sum }_{j=1}^{k}\frac{{x}_{{i}_{j}}}{1+{x}_{{i}_{j}}},\phantom{\rule{2.77695pt}{0ex}}k=1,...,n,$
(4)

and established the following results by using Theorem A.

Theorem D. For k = 1,..., n, F n (x , k) is an Schur-concave function on${ℝ}_{+}^{n}$.

In this note, by properties of the Schur-convex function, we simply prove Theorems B, C and D uniform.

## 2. New proofs three theorems

To prove the above three theorems, we need the following lemmas.

Lemma 1. [[1], p. 67], [2]If φ is symmetric and convex (concave) on symmetric convex set Ω, then φ is Schur-convex (Schur-concave) on Ω.

Lemma 2. [[1], p. 73],[2]Let Ω n, φ: Ω → +. Then lnφ is Schur-convex (Schur-concave) if and only if φ is Schur-convex (Schur-concave).

Lemma 3. [[1], p. 446], [2]Let Ω nbe open convex set, φ : Ω → . For x, y Ω, defined one variable function g(t) = φ (t x + (1 - t)y ) on interval (0, 1). Then φ is convex (concave) on Ω if and only if g is convex (concave) on (0, 1) for all x,y Ω.

Lemma 4. Let x= (x1,..., x m ) and y = (y1, ..., y m ) m. Then the following functions are concave on (0,1).

(i) $f\left(t\right)=\text{ln}{\sum }_{j=1}^{m}\left(t{x}_{j}+\left(1-t\right){y}_{j}\right)-\text{ln}{\sum }_{j=1}^{m}\left(1+t{x}_{j}+\left(1-t\right){y}_{j}\right)$,

(ii) $g\left(t\right)=\text{ln}{\sum }_{j=1}^{m}{\left(t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{1/m}$,

(iii) $h\left(t\right)=\frac{1}{m}\text{ln}\psi \left(t\right)$, where

$\psi \left(t\right)=\sum _{j=1}^{m}\frac{t{x}_{j}+\left(1-t\right){y}_{j}}{1+t{x}_{j}+\left(1-t\right){y}_{j}}.$

Proof. (i) Directly calculating yields

${f}^{\prime }\left(t\right)=\sum _{j=1}^{m}\left({x}_{j}-{y}_{j}\right)\left[\frac{1}{t{x}_{j}+\left(1-t\right){y}_{j}}-\frac{1}{1+t{x}_{j}+\left(1-t\right){y}_{j}}\right]$

and

$\begin{array}{c}{f}^{″}\left(t\right)=-\sum _{j=1}^{m}{\left({x}_{j}-{y}_{j}\right)}^{2}\left[\frac{1}{{\left(t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{2}}-\frac{1}{{\left(1+t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{2}}\right]\\ =-\sum _{j=1}^{m}{\left({x}_{j}-{y}_{j}\right)}^{2}\frac{1+2t{x}_{j}+2\left(1-t\right){y}_{j}}{{\left(t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{2}{\left(1+t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{2}}.\end{array}$

Since f''(t) ≤ 0, f(t) is concave on (0,1).

(ii) Directly calculating yields

${g}^{\prime }\left(t\right)=\frac{\frac{1}{m}{\sum }_{j=1}^{m}{\left({x}_{j}-{y}_{j}\right)}^{\frac{1}{m}-1}}{{\sum }_{j=1}^{m}{\left(t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{1/m}}$

and

${g}^{″}\left(t\right)=-\frac{{\left[\frac{1}{m}{\sum }_{j=1}^{m}{\left({x}_{j}-{y}_{j}\right)}^{\frac{1}{m}-1}\right]}^{2}}{{\sum }_{j=1}^{m}{\left(t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{2/m}}.$

Since g''(t) ≤ 0, f(t) is concave on (0,1)

(iii) By computing,

$\begin{array}{c}{h}^{\prime }\left(t\right)=\frac{1}{m}\frac{{\psi }^{\prime }\left(t\right)}{\psi \left(t\right)},\\ {h}^{″}\left(t\right)=\frac{1}{m}\frac{{\psi }^{″}\left(t\right)\psi \left(t\right)-{\left({\psi }^{\prime }\left(t\right)\right)}^{2}}{{\psi }^{2}\left(t\right)},\end{array}$

where

${\psi }^{\prime }\left(t\right)=\sum _{j=1}^{m}\frac{{x}_{j}-{y}_{j}}{{\left(1+t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{2}}$

and

${\psi }^{″}\left(t\right)=-\sum _{j=1}^{m}\frac{2{\left({x}_{j}-{y}_{j}\right)}^{2}}{{\left(1+t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{3}}.$

Thus,

$\begin{array}{ll}\hfill {\psi }^{″}\left(t\right)\psi \left(t\right)-{\left({\psi }^{\prime }\left(t\right)\right)}^{2}& =-\sum _{j=1}^{m}\frac{2{\left({x}_{j}-{y}_{j}\right)}^{2}}{{\left(1+t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{3}}\sum _{j=1}^{m}\frac{t{x}_{j}+\left(1-t\right){y}_{j}}{1+t{x}_{j}+\left(1-t\right){y}_{j}}\phantom{\rule{2em}{0ex}}\\ -{\left[\sum _{j=1}^{m}\frac{{x}_{j}-{y}_{j}}{{\left(1+t{x}_{j}+\left(1-t\right){y}_{j}\right)}^{2}}\right]}^{2}\le 0,\phantom{\rule{2em}{0ex}}\end{array}$

and then h'' (t) ≤ 0, so f(t) is concave on (0,1).

The proof of Lemma 4 is completed.

Proof of Theorem A: For any 1 ≤ i1 < · · · < i k n, by Lemma 3 and Lemma 4(i), it follows that $\text{ln}{\sum }_{j=1}^{k}{x}_{{i}_{j}}-\text{ln}{\sum }_{j=1}^{k}\left(1+{x}_{{i}_{j}}\right)$ is concave on ${ℝ}_{+}^{n}$, and then $\text{ln}{F}_{n}\left(x,k\right)={\prod }_{1\le {i}_{1}<\cdots <{i}_{k}\le n}\left(\text{ln}{\sum }_{j=1}^{k}{x}_{{i}_{j}}-\text{ln}{\sum }_{j=1}^{k}\left(1+{x}_{{i}_{j}}\right)\right)$ is concave on ${ℝ}_{+}^{n}$. Furthermore, it is clear that ln F n (x, k) is symmetric on ${ℝ}_{+}^{n}$, by Lemma 1, it follows that ln F n ( x, k) is concave on ${ℝ}_{+}^{n}$, and then from Lemma 2 we conclude that F n (x , k) is also concave on ${ℝ}_{+}^{n}$.

The proof of Theorem A is completed.

Similar to the proof of Theorem A, by Lemma 4 (ii) and Lemma 4 (iii), we can prove Theorems B and C, respectively. Omitted detailed process.

## References

1. Marshall AW, Olkin I: Inequalities:theory of majorization and its application. Academies Press, New York; 1979.

2. Wang B-Y: Foundations of majorization inequalities. Beijing Normal Univ. Press, Beijing, China, (Chinese); 1990.

3. Chu Y-M, Xia W-F, Zhao T-H: Some properties for a class of symmetric functions and applications. J Math Inequal 2011, 5(1):1–11.

4. Jiang W-D: Some properties of dual form of the Hamy's symmetric function. J Math Inequal 2007, 1(1):117–125.

5. Xia W-F, Chu Y-M: Schur-convexity for a class of symmetric functions and its applications. J Inequal Appl 2009., 15: vol. 2009, Article ID 493759

## Acknowledgements

Shi was supported in part by the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201111417006). This article was typeset by using $\mathcal{A}\mathcal{M}\mathcal{S}-LATEX$.

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Correspondence to Huan-Nan Shi.

### Competing interests

The authors declare that they have no competing interests.

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Shi, HN., Zhang, J. & Gu, C. New proofs of Schur-concavity for a class of symmetric functions. J Inequal Appl 2012, 12 (2012). https://doi.org/10.1186/1029-242X-2012-12

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• DOI: https://doi.org/10.1186/1029-242X-2012-12

### Keywords

• majorization
• Schur-concavity
• inequality
• symmetric functions
• concave functions