# Schur convexity for the ratios of the Hamy and generalized Hamy symmetric functions

• Wei-Mao Qian
Open Access
Research

## Abstract

In this paper, we present the Schur convexity and monotonicity properties for the ratios of the Hamy and generalized Hamy symmetric functions and establish some analytic inequalities. The achieved results is inspired by the paper of Hara et al. [J. Inequal. Appl. 2, 387-395, (1998)], and the methods from Guan [Math. Inequal. Appl. 9, 797-805, (2006)]. The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal.

2010 Mathematics Subject Classification: Primary 05E05; Secondary 26D20.

## Keywords

Hamy symmetric function generalized Hamy symmetric function Schur convex Schur concave

## 1 Introduction

Throughout this paper, we denote . For , the Hamy symmetric function [1] is defined as
(1.1)

where r is an integer and 1 ≤ rn.

The generalized Hamy symmetric function was introduced by Guan [2] as follows
(1.2)

where r is a positive integer.

In [2], Guan proved that both F n (x,r) and are Schur concave in . The main of this paper is to investigate the Schur convexity for the functions and and establish some analytic inequalities by use of the theory of majorization.

For convenience of readers, we recall some definitions as follows, which can be found in many references, such as [3].

Definition 1.1. The n-tuple x is said to be majorized by the n-tuple y (in symbols xy), if

where 1 ≤ kn - 1, and x[i]denotes the i th largest component of x.

Definition 1.2. Let E n be a set. A real-valued function F : E is said to be Schur convex on E if F(x) ≤ F(y) for each pair of n-tuples x = (x1,..., x n ) and y = (y1,..., y n ) in E, such that xy. F is said to be Schur concave if -F is Schur convex.

The theory of Schur convexity is one of the most important theories in the fields of inequalities. It can be used in combinatorial optimization [4], isoperimetric problems for polytopes [5], theory of statistical experiments [6], graphs and matrices [7], gamma functions [8], reliability and availability [9], optimal designs [10] and other related fields.

Our aim in what follows is to prove the following results.

Theorem 1.1. Let is an integer, then the function is Schur concave in and increasing with respect to x i (i= 1,2, ...,n).

Theorem 1.2. Let is an integer, then the function is Schur concave in and increasing with respect to x i (i= 1,2,... n).

Corollary 1.1. If and that cs, then
and

where are the arithmetic and geo-metric means of x, respectively.

Corollary 1.2. If and that cs, then
and

## 2 Lemmas

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

Lemma 2.1 (see [3]). Let E n be a symmetric convex set with nonempty interior intE and φ : E be a continuous symmetric function. If φ is differentiable on intE, then φ is Schur convex (or Schur concave, respectively) on E if and only if

for all i,j = 1,2,...,n and x = (x1,...,x n ) ∈ intE.

The r th elementary symmetric function (see [11]) is defined as
(2.1)

where 1 ≤ rn is a positive integer, and E n (x, 0) = 1.

By (2.1) and simple computations, we have the following lemma.

Lemma 2.2. Let , if
Then,
(2.2)

Lemma 2.3 (see [11]). Let is an integer and 1 ≤ rn - 1.

Then,
(2.3)
Another important symmetric function is the complete symmetric function (see [3]), which is defined by

where i1, i2,..., i n are non-negative integer, r ∈ {1, 2,...} and C0(x) = 1.

Lemma 2.4 (see [12]). Let x i > 0, i = 1, 2,..., n, and .

Then,
Lemma 2.5 (see [13]). If , then
Lemma 2.6 (see [14]). If and cs, then
1. (1)

,

(2).

## 3 Proof of Theorems

Proof of Theorem 1.1. It is obvious that ϕ r (x) is symmetric and has continuous partial derivatives in . By Lemma 2.1, we only need to prove that
(3.1)
For any fixed 2 ≤ rn, let and , we have
Differentiating ϕ r (x) with respect to x1 yields
(3.2)
Using Lemma 2.2 repeatedly, we get
(3.3)
Equations (3.2) and (3.3) lead to
(3.4)
where
and
Similarly, we can deduce that
(3.5)
From (3.4) and (3.5), one has
(3.6)
It follows from (3.3) and Lemma 2.3 that

Similarly, we can get B > 0.

It follows from the function is decreasing in (0, +∞) that
(3.7)

Therefore, inequality (3.1) follows from (3.6) and (3.7) together with A > 0 and B > 0.

Next, we prove that is increasing with respect to x i (i= 1,2,...,n).

By the symmetry of ϕ r (x) with respect to x i (i = 1, 2,..., n), we only need to prove that

which can be derived directly from A > 0 and B > 0 together with Equation (3.4).

Proof of Theorem 1.2. It is obvious that is symmetric and has continuous partial derivatives in . By Lemma 2.1, we only need to prove that
(3.8)
For any fixed 2 ≤ rn, let and . Then,
(3.9)
Differentiating with respect to x1, we have
(3.10)
It follows from Lemma 2.4 that
(3.11)
Equations (3.10) and (3.11) lead to
(3.12)
Similarly, we have
(3.13)
From (3.12) and (3.13), one has
(3.14)
By Lemma 2.5, we know that
(3.15)
The monotonicity of the function in (0, +∞) leads to the conclusion that
(3.16)

Therefore, inequality (3.8) follows from (3.14)-(3.16).

Next, we prove that is increasing with respect to x i (i= 1,2,...,n).

From (3.12) and (3.15), we clearly see that
(3.17)

Inequality (3.17) implies that is increasing with respect to x1, then from the symmetry of with respect to x i (i = 1, 2,..., n) we know that is increasing with respect to each x i (i = 1, 2,..., n).

Proof of Corollary 1.1. By Theorem 1.1 and Lemma 2.6, we have and which imply Corollary 1.1.

Remark 1. Let , then
(3.18)

where (1 - x) = (1 - x1, 1 - x2,... , 1 - x n ), commonly referred to as Ky Fan inequality (see [15]), which has attracted the attention of a considerable number of mathematicians (see [16, 17, 18, 19, 20]).

Letting and taking c = 1 in Corollary 1.1, we get
(3.19)

It is obvious that inequality (3.19) can be called Ky Fan-type inequality.

Remark 2. Let x i > 0, i = 1, 2,..., n, the following inequalities
and

are the well-known Weierstrass inequalities (see [11]).

Taking c = s = 1 in Corollary 1.1, one has
and

It is obvious that our inequalities can be called Weierstrass-type inequalities.

Proof of Corollary 1.2. By Theorem 1.2 and Lemma 2.6, we have and , which imply Corollary 1.2.

## Notes

### Acknowledgements

This work was supported by NSF of China under grant No. 11071069.

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