The Schur Harmonic Convexity of the Hamy Symmetric Function and Its Applications

Abstract

We prove that the Hamy symmetric function is Schur harmonic convex for . As its applications, some analytic inequalities including the well-known Weierstrass inequalities are obtained.

1. Introduction

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

For , and , we denote by

(1.1)

For , the Hamy symmetric function [1–3] was defined as

(1.2)

Corresponding to this is the r th order Hamy mean

(1.3)

where . Hara et al. [1] established the following refinement of the classical arithmetic and geometric means inequality:

(1.4)

Here and denote the classical arithmetic and geometric means, respectively.

The paper [4] by Ku et al. contains some interesting inequalities including the fact that is log-concave, the more results can also be found in the book [5] by Bullen. In [2], the Schur convexity of Hamy's symmetric function and its generalization were discussed. In [3] , Jiang defined the dual form of the Hamy symmetric function as follows:

(1.5)

discussed the Schur concavity Schur convexity of , and established some analytic inequalities.

The main purpose of this paper is to investigate the Schur harmonic convexity of the Hamy symmetric function . Some analytic inequalities including Weierstrass inequalities are established.

2. Definitions and Lemmas

Schur convexity was introduced by Schur in 1923 [6], and it has many important applications in analytic inequalities [7–12], linear regression [13], graphs and matrices [14], combinatorial optimization [15], information-theoretic topics [16], Gamma functions [17], stochastic orderings [18], reliability [19], and other related fields.

For convenience of readers, we recall some definitions as follows.

Definition 2.1.

A set is called a convex set if whenever . A set is called a harmonic convex set if whenever .

It is easy to see that is a harmonic convex set if and only if is a convex set.

Definition 2.2.

Let be a convex set a function is said to be convex on if for all . Moreover, is called a concave function if is a convex function.

Definition 2.3.

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

Definitions 2.2 and 2.3 have the following consequences.

Fact A.

If is a harmonic convex set and is a harmonic convex function, then

(2.1)

is a concave function. Conversely, if is a convex set and is a convex function, then

(2.2)

is a harmonic concave function.

Definition 2.4.

Let be a set a function is called a Schur convex function on if

(2.3)

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

(2.4)

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

Definition 2.5.

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

(2.5)

for each pair of and in , such that .

Definitions 2.4 and 2.5 have the following consequences.

Fact B.

Let be a set, and , then is a Schur harmonic convex (or concave, resp.) function on if and only if is a Schur concave (or convex, resp.) function on .

The notion of generalized convex function was first introduced by Aczél in [20]. Later, many authors established inequalities by using harmonic convex function theory [21–28]. Recently, Anderson et al. [29] discussed an attractive class of inequalities, which arise from the notation of harmonic convex functions.

The following well-known result was proved by Marshall and Olkin [6].

Theorem 2 A.

Let be a symmetric convex set with nonempty interior , and let be a continuous symmetric function on . If is differentiable on , then is Schur convex (or concave, resp.) on if and only if

(2.6)

for all and . Here, is a symmetric set means that implies for any permutation matrix .

Remark 2.6.

Since is symmetric, the Schur's condition in Theorem , that is, (2.6) can be reduced to

(2.7)

The following Lemma 2.7 can easily be derived from Fact, Theorem and Remark 2.6 together with elementary computation.

Lemma 2.7.

Let be a symmetric harmonic convex set with nonempty interior , and let be a continuous symmetry function on . If is differentiable on , then is Schur harmonic convex (or concave, resp.) on if and only if

(2.8)

for all .

Next we introduce two lemmas, which are used in Sections 3 and 4.

Lemma 2.8 ([5, page 234]).

For , if th r -th order symmetric function is defined as

(2.9)

then

(2.10)

Lemma 2.9 ([2, Lemma  2.2]).

Suppose that and . If , then

(2.11)

3. Main Result

In this section, we give and prove the main result of this paper.

Theorem 3.1.

The Hamy symmetric function is Schur harmonic convex in .

Proof.

By Lemma 2.7, we only need to prove that

(3.1)

To prove (3.1), we consider the following possible cases for .

Case 1 ().

Then (1.2) leads to , and (3.1) is clearly true.

Case 2 ().

Then (1.2) leads to the following identity:

(3.2)

and therefore, (3.1) follows from (3.2).

Case 3 ().

Then (1.2) leads to

(3.3)

Simple computation yields

(3.4)

From (3.4) we get

(3.5)

Therefore, (3.1) follows from (3.5) and the fact that is increasing in .

Case 4 ().

Fix and let and We have the following identity:

(3.6)

Differentiating (3.6) with respect to and , respectively, and using Lemma 2.8, we get

(3.7)

From (3.7) we obtain

(3.8)

Therefore, (3.1) follows from (3.8) and the fact that is increasing in .

4. Applications

In this section, making use of our main result, we give some inequalities.

Theorem 4.1.

Suppose that with . If and , then

(4.1)

Proof.

The proof follows from Theorem 3.1 and Lemma 2.9 together with (1.2).

If taking and in Theorem 4.1, respectively, then we have the following corollaries.

Corollary 4.2.

Suppose that with . If , then

(4.2)

Corollary 4.3.

Suppose that with . If , then

(4.3)

Taking in Corollaries 4.2 and 4.3, respectively, we get the following.

Corollary 4.4.

If and , then

(4.4)

Corollary 4.5 (Weierstrass inequalities [30, Page 260]).

If , and , then

(4.5)

Theorem 4.6.

If and , then

(4.6)

Proof.

Let , and be the -tuple, then obviously

(4.7)

Therefore, Theorem 4.6 follows from Theorem 3.1, (4.7) ,and (1.2).

Theorem 4.7.

Let be an -dimensional simplex in -dimensional Euclidean space , and be the set of vertices. Let be an arbitrary point in the interior of A. If is the intersection point of the extension line of and the -dimensional hyperplane opposite to the point , and , then one has

(4.8)

Proof.

It is easy to see that

(4.9)

(4.9) implies that

(4.10)

Therefore, Theorem 4.7 follows from Theorem 3.1, (4.10), and (1.2).