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# On Schur Convexity of Some Symmetric Functions

*Journal of Inequalities and Applications***volume 2010**, Article number: 543250 (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.

## 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.

Let be a set, and a real-valued function on is called a Schur convex function if

for each pair of -tuples and in with , that is,

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

Definition 1.2.

Let be a set, and a function is called a Schur multiplicatively convex function on if

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

Definition 1.3.

Let be a set. A function is called a Schur harmonic convex (or Schur harmonic concave, resp.) function on if

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]).

Let be a continuous symmetric function. If is differentiable in , then is Schur convex in if and only if

for all

Lemma 1.5 (see [7]).

Let be a continuous symmetric function. If is differentiable in , then is Schur multiplicatively convex in if and only if

for all

Lemma 1.6 (see [10]).

Let be a continuous symmetric function. If is differentiable in , then is Schur harmonic convex in if and only if

for all

Lemma 1.7 (see [12]).

Let and . If , then

Lemma 1.8 (see [12]).

Let and . If , then

Lemma 1.9 (see [13]).

Suppose that and If then

## 2. Main Result

Theorem 2.1.

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

Proof.

According to Lemmas 1.4–1.6 we only need to prove that

for all and

We divided the proof into seven cases.

Case 1.

If and , then (1.7) leads to

Case 2.

If and , then (1.7) yields

Case 3.

If , , and , then from (1.7) we clearly see that

Case 4.

If , , and , then (1.7) implies that

Case 5.

If , and , then (1.7) leads to

Case 6.

If , , and , then (1.7) yields

Case 7.

If , , and , then (1.7) implies

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.

If , , and , then

Theorem 3.2.

If , , , and , then

Proof.

Theorem 3.2 follows from Theorem 2.1 and the fact that

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

Corollary 3.3.

If with and then

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

Corollary 3.4.

If with and then

Remark 3.5.

If we take in Corollaries 3.3 and 3.4, then we have

for and

Theorem 3.6.

If , then

Proof.

Theorem 3.6 follows from Theorem 2.1 and (1.7) together with the fact that

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

Corollary 3.7.

If , then

Theorem 3.8.

Let be an -dimensional simplex in and let be an arbitrary point in the interior of . If is the intersection point of straight line and hyperplane , then

Proof.

It is easy to see that and , and these identities imply that

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

Theorem 3.9.

Suppose that is a complex matrix, and are the eigenvalues of . If is a positive definite Hermitian matrix, then

Proof.

We clearly see that and

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 .

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## Acknowledgments

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

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### Keywords

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