Abstract and Applied Analysis

Abstract and Applied Analysis / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 830163 | 16 pages | https://doi.org/10.1155/2010/830163

Necessary and Sufficient Conditions for Schur Geometrical Convexity of the Four-Parameter Homogeneous Means

Academic Editor: Roman Šimon Hilscher
Received20 Feb 2010
Revised17 Mar 2010
Accepted23 Mar 2010
Published22 Apr 2010

Abstract

The necessary and sufficient conditions for Schur geometrical convexity of the four-parameter means are given. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means.

1. Introduction and Main Result

Let and . For the Stolarsky means are defined as and (see [1]), where are the logarithmic mean and identric (exponential) mean of positive numbers and , respectively.

Another two-parameter family of means was introduced by Gini in [2]. That are defined as

Stolarsky and Gini means both are contained in the so-called four-parameter means [3], which are defined as follows.

Definition 1.1. Let with and . Then the four-parameter homogeneous means denoted by are defined as follows: or
If then are defined as their corresponding limits, for example: where denote logarithmic mean and identric (exponential) mean, respectively, .

The Schur convexity of and on with respect to was investigated by Qi et al. [4], Shi et al. [5], Li and Shi [6], and Chu and Zhang [7]. Until now, they have been perfectly solved by Chu and Zhang [7], Wang and Zhang [8], respectively. Recently, Chu and Xia also proved the same result as Wang and Zhang [9].

The Schur convexity of and on and with respect to was investigated by Qi [10] and Sándor [11], respectively. Now Schur convexity of a four-parameter homogeneous means family containing Stolarsky and Gini means on with respect to has been perfectly solved by Yang [12].

The Schur geometrical convexity was introduced by Zhang [13]. In [8, 14], Wand and Zhang proved that is Schur geometrically convex (Schur geometrically concave) on with respect to if . Chu et al. [15] pointed out that this conclusion is also true for . Shi et al. [5, 16], Li and Shi [6], and Gu and Shi [17] also obtained similar results.

The purpose of this paper is to present the necessary and sufficient conditions for Schur geometrical convexity of the four-parameter homogeneous means. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means with respect to .

Our main result is as follows.

Theorem 1.2. For fixed the four-parameter homogeneous means are Schur geometrically convex (Schur geometrically concave) on with respect to if and only if .

2. Definitions and Lemmas

Definition 2.1 (see [18, 19]). Let and .(i) is said to by majorized by (in symbol ) if where and are rearrangements of and in a decreasing order.(ii) means for all . Let . The function is said to be increasing if implies . is said to be decreasing if and only if is increasing.(iii) is called a convex set if for all and , where with .(iv)Let be a set with nonempty interior. Then is said to be Schur convex if on implies . is said to be Schur concave if is Schur convex.

Definition 2.2 (see [13, 20]). Let and . Denote (i) is called a geometrically convex set if for all and , where with .(ii)Let be a set with nonempty interior. Then function is said to be Schur geometrically convex on if on implies . is said to be Schur geometrically concave if is Schur geometrically convex.

Definition 2.3 (see [18]). (i) is called symmetric set if implies for every permutation matrix .
(ii) The function is called symmetric if for every permutation matrix , for all .

Lemma 2.4 (see [18, 19]). Let be a symmetric set with nonempty interior and be continuous on and differentiable in . Then is Schur convex (Schur concave) on if and only if is symmetric on and holds for any .

Lemma 2.5 (see [13, Theorem 1.4, page 108]). Let be a symmetric set with a nonempty interior geometrically convex set . Let be continuous on and differentiable in . Then is Schur geometrically convex (Schur geometrically concave) on if and only if is symmetric on and holds for any .

3. Schur Geometrical Convexity of Two-Parameter Homogeneous Functions

The more general form of two-parameter homogeneous means is the so-called two-parameter homogenous functions first introduced by Yang [21]. For conveniences, we record it as follows.

Definition 3.1. Assume that : is -order homogeneous, continuous and exists first partial derivatives and , .
If for and for all , then define where and denote first-order partial derivatives with respect to first and second component of ,respectively.
If for all , then define further Since is a homogeneous function, is also one and called a homogeneous function with parameters and and simply denoted by or sometimes.

Concerning the monotonicity and log-convexity of two-parameter homogeneous functions, there have been some literatures such as [3, 21, 22], which yield some new and interesting inequalities for means.

The two-parameter homogeneous functions have some well properties (see [2123]) such as the following lemma.

Lemma 3.2 (see [23]). Let be a homogenous and differentiable function and Then we have

Next we give another property.

Lemma 3.3. Let be a homogenous and -time differentiable function. Then .

Proof. Since has continuous partial derivatives of order with respect to on , the integrand in (3.6) has continuous partial derivatives of order with respect to on , that is .

For the Schur geometrical convexity, we have the following result.

Theorem 3.4. Assume that : is a symmetric, -order homogeneous, continuous, and three-time differentiable function. If for any with then is Schur geometrically convex on with respect to if and only if and Schur geometrically concave if and only if .

Proof. () In the case of . We have Some simple partial derivative computations yield hence, where It is easy to verify that is even on . In fact, since is -order homogeneous and symmetric, for arbitrary , we have Thus, Let , . Then Note and both are -order homogeneous with respect to and , then and then Therefore, By the mean values theorem, there is a between and such that where , Thus we have By Lemma 2.5, our required result is derived immediately.
() In the case of . By Lemma 3.3 together with (3.10) and (3.17), we have where , . Hence we have By Lemma 2.5, the required result holds.
() In the case of . By Lemma 3.3 and (3.20), we have However, where due to the symmetry of . Thus
Summarizing the above three cases, this proof of Theorem 3.4 is complete.

4. Proof of Main Result

Establishing the Theorem 3.4, we are in a position to prove main result.

Proof of Theorem 1.2. It follows from [3, Section 1], that where is symmetric with respect to and . From Lemma 3.3, it follows that . Thus we have () In the case of .
Simple partial derivative calculations yield Hence, where It is easy to check that is even and increasing (decreasing) on if . Indeed, With , then , and then can be written as Direct computation yields From it follows that if , that is, and if . Namely,
By the mean values theorem, there is a between and such that and then Using Theorem 3.4, for fixed with , the four-parameter homogeneous means are Schur geometrically convex on with respect to if and only if and Schur geometrically concave if and only if .
() In the case of , .
From (4.11) together with (4.4)–(4.6) and (4.19), there is a between and such that
() In the case of , .
Since is symmetric with respect to and , it follows from case 2 that
() In the case of .
From (4.11) together with (4.1)–(4.3), we have
() In the case of .
From (4.22) together with (4.7)–(4.9), we have But by (4.15) and some limit computations, we obtain which implies .
Summarizing the above five cases, our required results are derived.
This proof ends.

5. Other Corollaries

The four-parameter homogeneous means also contain many other two-parameter means, for instance, for the identric (exponential) mean defined by (1.3), its two-parameter means are defined as follows [21, Example 2.3]: where

By [3], we see that And then according to Theorem 1.2, we have the following corollary.

Corollary 5.1. For fixed , the two-parameter identric (exponential) means are Schur geometrically convex on with respect to if and only if and Schur geometrically concave if and only if .

As another example, for Heronian mean defined by its two-parameter means are defined as follows:

By [3], we see that And then according to Theorem 1.2, we have the following corollary.

Corollary 5.2. For fixed , the two-parameter Heronian means are Schur geometrically convex on with respect to if and only if and Schur geometrically concave if and only if .

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Copyright © 2010 Zhen-Hang Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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