Schur-Convexity for a Class of Symmetric Functions and Its Applications

Abstract

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

1. Introduction

Throughout this paper we use to denote the dimensional Euclidean space over the field of real numbers and . In particular, we use to denote

For the sake of convenience, we use the following notation system.

For , and , let

(1.1)

The notion of Schur convexity was first introduced by Schur in 1923 [1]. It has many important applications in analytic inequalities [2–7], combinatorial optimization [8], isoperimetric problem for polytopes [9], linear regression [10], graphs and matrices [11], gamma and digamma functions [12], reliability and availability [13], and other related fields. The following definition for Schur convex or concave can be found in [1, 3, 7] and the references therein.

Definition 1.1.

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

(1.2)

for each pair of -tuples and on , such that is majorized by ( in symbols ), that is,

(1.3)

where denotes the th largest component in . is called Schur concave if is Schur convex.

The notation of multiplicative convexity was first introduced by Montel [14]. The Schur multiplicative convexity was investigated by Niculescu [15], Guan [7], and Chu et al.[16].

Definition 1.2 (see [7, 16]).

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

(1.4)

for each pair of tuples and on , such that is logarithmically majorized by (in symbols ), that is,

(1.5)

However is called Schur multiplicatively concave if is Schur multiplicatively convex.

In paper [17], Anderson et al. discussed an attractive class of inequalities, which arise from the notion of harmonic convex functions. Here, we introduce the notion of Schur harmonic convexity.

Definition 1.3.

Let be a set. A real-valued function on is called a Schur harmonic convex function if

(1.6)

for each pair of tuples and on , such that . is called a Schur harmonic concave function on if (1.6) is reversed.

The main purpose of this paper is to discuss the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of the following symmetric function:

(1.7)

where , , and are positive integers. As applications, some inequalities are established by use of the theory of majorization.

2. Lemmas

In order to establish our main results we need several lemmas, which we present in this section.

The following lemma is so-called Schur's condition which is very useful for determining whether or not a given function is Schur convex or Schur concave.

Lemma 2.1 (see [6, 7, 18]).

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

(2.1)

for all and . Also is Schur concave if and only if (2.1) is reversed for all and . Here, is a symmetric function in meaning that for any and any permutation matrix .

Remark 2.2.

Since is symmetric, the Schur's condition in Lemma 2.1, that is, (2.1) can be reduced to

(2.2)

Lemma 2.3 (see [7, 16]).

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

(2.3)

for all . Also is Schur multiplicatively concave if and only if (2.3) is reversed.

Lemma 2.4.

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

(2.4)

for all . Also is Schur harmonic concave if and only if (2.4) is reversed.

Proof.

From Definitions 1.1 and 1.3, we clearly see the fact that is Schur harmonic convex if and only if is Schur concave.

This fact, Lemma 2.1 and Remark 2.2 together with elementary calculation imply that Lemma 2.4 is true.

Lemma 2.5 (see [5, 6]).

Let and . If , then

(2.5)

Lemma 2.6 (see [6]).

Let and . If , then

(2.6)

Lemma 2.7 (see [19]).

Suppose that and If then

(2.7)

3. Main Results

Theorem 3.1.

For the symmetric function is Schur concave in .

Proof.

By Lemma 2.1 and Remark 2.2, we only need to prove that

(3.1)

for all and

The proof is divided into four cases.

Case 1.

If , then (1.7) leads to

(3.2)

However (3.2) and elementary computation lead to

(3.3)

Case 2.

If and , then (1.7) yields

(3.4)

From (3.4) and elementary computation, we have

(3.5)

Case 3.

If and , then by (1.7) we have

(3.6)

Elementary computation and (3.6) yield

(3.7)

Case 4.

If and , then from (1.7), we have

(3.8)

(3.9)

Therefore, (3.1) follows from Cases 1–4 and the proof of Theorem 3.1 is completed.

For the Schur multiplicative convexity or concavity of , we have the following theorem

Theorem 3.2.

It holds that is Schur multiplicatively concave in

Proof.

According to Lemma 2.3 we only need to prove that

(3.10)

for all and Then proof is divided into four cases.

Case 1.

If , then (3.2) leads to

(3.11)

Case 2.

If , then (3.4) yields

(3.12)

Case 3.

If and , then (3.6) implies

(3.13)

Case 4.

If and , then from (3.8) we have

(3.14)

Therefore, Theorem 3.2 follows from (3.10) and Cases 1–4.

Remark 3.3.

From (3.11) and (3.12) we know that is Schur multiplicatively concave in and is Schur multiplicatively convex in .

Theorem 3.4.

For the symmetric function is Schur harmonic convex in .

Proof.

According to Lemma 2.4 we only need to prove that

(3.15)

for all and

The proof is divided into four cases.

Case 1.

If , then from (3.2) we have

(3.16)

Case 2.

If and , then (3.4) leads to

(3.17)

Case 3.

If and , then (3.6) yields

(3.18)

Case 4.

If and , then (3.8) implies

(3.19)

Therefore, (3.15) follows from Cases 1–4 and the proof of Theorem 3.4 is completed.

4. Applications

In this section, we establish some inequalities by use of Theorems 3.1, 3.2 and 3.4 and the theory of majorization.

Theorem 4.1.

If , ,, and , then

(1) for

(2) for

(3) for

(4) for

(5) for

(6) for

(7) for

(8) for

Proof.

Theorem 4.1 follows from Theorem 3.1, Theorem 3.4 and Lemmas 2.5–2.7 together with the fact that

(4.1)

Theorem 4.2.

If , , and , then

(4.2)

Proof.

We clearly see that

(4.3)

Therefore, Theorem 4.2 follows from (4.3) and Theorem 3.1 together with (1.7), and Theorem 4.2 follows from (4.3) and Theorem 3.4 together with (1.7).

If we take in Theorem 4.2 and in Theorem 4.2, respectively, then we have the following corollary.

Corollary 4.3.

If and , then

(4.4)

Remark 4.4.

If we take in Corollary 4.3, then we obtain the Weierstrass inequality: (see [20, page 260])

(4.5)

Theorem 4.5.

If and , then

(4.6)

Proof.

We clearly see that

(4.7)

Therefore, Theorem 4.5 follows from (4.7) and Theorem 3.4 together with (1.7), and Theorem 4.5 follows from (4.7) and Theorem 3.1 together with (1.7).

If we take and in Theorem 4.5, respectively, then we get the following corollary.

Corollary 4.6.

If , then

(4.8)

Theorem 4.7.

If and , then

(4.9)

Proof.

We clearly see that

(4.10)

Therefore, Theorem 4.7 follows from (4.10), Theorem 3.2, and (1.7).

If we take and in Theorem 4.7, respectively, then we get the following corollary.

Corollary 4.8.

If , then

(4.11)

Remark 4.9.

From Remark 3.3 and (4.10) together with (1.7) we clearly see that inequality in Corollary 4.8 is reversed for and inequality in Corollary 4.8 is true for .

Theorem 4.10.

If then

(4.12)

Proof.

Theorem 4.10 follows from Theorems 3.1, 3.4, and (1.7) together with the fact that

(4.13)

Theorem 4.11.

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 for one has

(4.14)

Proof.

It is easy to see that and , these identical equations imply

(4.15)

Therefore, Theorem 4.11 follows from (4.15), Theorems 3.1, 3.4, and (1.7).

Remark 4.12.

Mitrinovi et al. [21, pages 473–479] established a series of inequalities for and , . Obvious, our inequalities in Theorem 4.11 are different from theirs.

Theorem 4.13.

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

(4.16)

Proof.

We clearly see that and then we have

(4.17)

Therefore, Theorem 4.13 and follows from (4.17), Theorems 3.1, 3.4, and (1.7).

It is easy to see that are the eigenvalues of matrix and , then we get

(4.18)

Therefore, Theorem 4.13 follows from (4.18), Theorem 3.2, and (1.7).

It is not difficult to verify that

(4.19)

Therefore, Theorem 4.13 follows from (4.19), and Theorem 3.2 together with (1.7).

A result due to Weyl [22] gives

(4.20)

From (4.20), we clearly see that

(4.21)

Therefore, Theorem 4.13 follows from (4.21), Theorem 3.2, and (1.7).