On Schur Convexity of Some Symmetric Functions

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

(1.1)

If then we denote by

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

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

(1.8)

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

(1.9)

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

(1.10)

for all

Lemma 1.7 (see [12]).

Let and . If , then

(1.11)

Lemma 1.8 (see [12]).

Let and . If , then

(1.12)

Lemma 1.9 (see [13]).

Suppose that and If then

(1.13)

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

(2.1)

(2.2)

(2.3)

for all and

We divided the proof into seven cases.

Case 1.

If and , then (1.7) leads to

(2.4)

(2.5)

(2.6)

(2.7)

Case 2.

If and , then (1.7) yields

(2.8)

(2.9)

(2.10)

(2.11)

Case 3.

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

(2.12)

(2.13)

(2.14)

(2.15)

Case 4.

If , , and , then (1.7) implies that

(2.16)

(2.17)

(2.18)

(2.19)

Case 5.

If , and , then (1.7) leads to

(2.20)

(2.21)

(2.22)

(2.23)

Case 6.

If , , and , then (1.7) yields

(2.24)

(2.25)

(2.26)

(2.27)

Case 7.

If , , and , then (1.7) implies

(2.28)

(2.29)

(2.30)

(2.31)

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

(3.1)

Theorem 3.2.

If , , , and , then

(3.2)

Proof.

Theorem 3.2 follows from Theorem 2.1 and the fact that

(3.3)

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

(3.4)

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

(3.5)

Remark 3.5.

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

(3.6)

for and

Theorem 3.6.

If , then

(3.7)

Proof.

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

(3.8)

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

Corollary 3.7.

If , then

(3.9)

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

(3.10)

Proof.

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

(3.11)

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

(3.12)

Proof.

We clearly see that and

(3.13)

(3.14)

(3.15)

(3.16)

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 .