# On Schur Convexity of Some Symmetric Functions

- Wei-Feng Xia
^{1}and - Yu-Ming Chu
^{2}Email author

**2010**:543250

https://doi.org/10.1155/2010/543250

© W.-F. Xia and Y.-M. Chu 2010

**Received: **20 November 2009

**Accepted: **3 March 2010

**Published: **8 March 2010

## Abstract

## Keywords

## 1. Introduction

Throughout this paper, we use the following notation system.

Next we introduce some definitions and well-known results.

Definition 1.1.

where denotes the th largest component of . A function is called Schur concave if is Schur convex.

Definition 1.2.

for each pair of -tuples and in with . is called Schur multiplicatively concave if is Schur multiplicatively convex.

Definition 1.3.

for each pair of -tuples and in with .

Schur convexity was introduced by Schur [1] in 1923 and it has many applications in analytic inequalities [2, 3], extended mean values [4, 5], graphs and matrices [6], and other related fields. Recently, the Schur multiplicative convexity was investigated in [7–9] and the Schur hamonic convexity was discussed in [10].

For and , the symmetric function is defined as

The aim of this article is to discuss the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of the symmetric function .

Lemma 1.4 (see [11]).

Lemma 1.5 (see [7]).

Lemma 1.6 (see [10]).

Lemma 1.7 (see [12]).

Lemma 1.8 (see [12]).

Lemma 1.9 (see [13]).

## 2. Main Result

Theorem 2.1.

The symmetric function is Schur convex, Schur multiplicatively convex, and Schur harmonic convex in for all

Proof.

We divided the proof into seven cases.

Case 1.

Case 2.

Case 3.

Case 4.

Case 5.

Case 6.

Case 7.

Therefore, inequality (2.1) follows from inequalities (2.5), (2.9), (2.13), (2.17), (2.21), (2.25), and (2.29), inequality (2.2) follows from inequalities (2.6), (2.10), (2.14),(2.18), (2.22), (2.26), and (2.30), and inequality (2.3) follows from inequalities (2.7), (2.11), (2.15), (2.19), (2.23), (2.27), and (2.31).

## 3. Applications

In this section, we establish several inequalities by use of Theorem 2.1 and the theory of majorization.

It follows from Lemmas 1.7, 1.8, 1.9, and Theorem 2.1 that Theorem 3.1 is obvious.

Theorem 3.1.

Theorem 3.2.

Proof.

If we take and in Theorem 3.1(3) and Theorem 3.2, respectively, then we get the following.

Corollary 3.3.

If we take and in Theorem 3.1(3) and Theorem 3.2, respectively, then one gets the following.

Corollary 3.4.

Remark 3.5.

Theorem 3.6.

Proof.

If we take in Theorem 3.6, then we have the following.

Corollary 3.7.

Theorem 3.8.

Proof.

Therefore, Theorem 3.8 follows from Theorem 2.1 and (1.7) together with (3.11).

Theorem 3.9.

Proof.

Therefore, Theorem 3.9(1) follows from (1.7), (3.13), and the Schur convexity of , Theorems 3.9(2) and (3) follow from (3.14) and (3.15) together with the Schur multiplitively convexity of , and Theorem 3.9(4) follows from (3.16) and the Schur harmonic convexity of .

## Declarations

### Acknowledgments

The research was supported by NSF of China (no. 60850005) and the NSF of Zhejiang Province (nos. D7080080, Y7080185, and Y607128).

## Authors’ Affiliations

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