- Research Article
- Open Access

# 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

For and , the symmetric function is defined as , where are positive integers. In this paper, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of are discussed. As consequences, several inequalities are established by use of the theory of majorization.

## Keywords

- Convex Function
- Intersection Point
- Arbitrary Point
- Symmetric Function
- Related Field

## 1. Introduction

Throughout this paper, we use the following notation system.

For , and , let

If then we denote by

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

where are positive integers.

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]).

for all

Lemma 1.5 (see [7]).

for all

Lemma 1.6 (see [10]).

for all

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.

for all and

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.

for and

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

## References

- Schur I: Über eine Klasse Von Mittelbildungen mit Anwendungen auf die Determinantentheorie.
*Sitzungsberichte der Berliner Mathematischen Gesellschaft*1923, 22: 9–20.MATHGoogle Scholar - Hardy GH, Littlewood JE, Pólya G: Some simple inequalities satisfied by convex functions.
*Messenger of Mathematics*1928/1929, 58: 145–152.MATHGoogle Scholar - Zhang X-M: Schur-convex functions and isoperimetric inequalities.
*Proceedings of the American Mathematical Society*1998, 126(2):461–470. 10.1090/S0002-9939-98-04151-3MathSciNetView ArticleMATHGoogle Scholar - Qi F, Sándor J, Dragomir SS, Sofo A: Notes on the Schur-convexity of the extended mean values.
*Taiwanese Journal of Mathematics*2005, 9(3):411–420.MathSciNetMATHGoogle Scholar - Chu Y, Zhang X: Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave.
*Journal of Mathematics of Kyoto University*2008, 48(1):229–238.MathSciNetMATHGoogle Scholar - Constantine GM: Schur convex functions on the spectra of graphs.
*Discrete Mathematics*1983, 45(2–3):181–188. 10.1016/0012-365X(83)90034-1MathSciNetView ArticleMATHGoogle Scholar - Chu Y, Zhang X, Wang G: The Schur geometrical convexity of the extended mean values.
*Journal of Convex Analysis*2008, 15(4):707–718.MathSciNetMATHGoogle Scholar - Guan K: A class of symmetric functions for multiplicatively convex function.
*Mathematical Inequalities & Applications*2007, 10(4):745–753.MathSciNetView ArticleMATHGoogle Scholar - Jiang W-D: Some properties of dual form of the Hamy's symmetric function.
*Journal of Mathematical Inequalities*2007, 1(1):117–125.MathSciNetView ArticleMATHGoogle Scholar - Chu Y, Lv Y: The Schur harmonic convexity of the Hamy symmetric function and its applications.
*Journal of Inequalities and Applications*2009, 2009:-10.Google Scholar - Marshall AW, Olkin I:
*Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and Engineering*.*Volume 143*. Academic Press, New York, NY, USA; 1979:xx+569.Google Scholar - Guan K: Schur-convexity of the complete symmetric function.
*Mathematical Inequalities & Applications*2006, 9(4):567–576.MathSciNetView ArticleMATHGoogle Scholar - Wu S: Generalization and sharpness of the power means inequality and their applications.
*Journal of Mathematical Analysis and Applications*2005, 312(2):637–652. 10.1016/j.jmaa.2005.03.050MathSciNetView ArticleMATHGoogle Scholar

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