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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

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 ), 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 . That are defined as

Stolarsky and Gini means both are contained in the so-called four-parameter means , 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. , Shi et al. , Li and Shi , and Chu and Zhang . Until now, they have been perfectly solved by Chu and Zhang , Wang and Zhang , respectively. Recently, Chu and Xia also proved the same result as Wang and Zhang .

The Schur convexity of and on and with respect to was investigated by Qi  and Sándor , 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 .

The Schur geometrical convexity was introduced by Zhang . In [8, 14], Wand and Zhang proved that is Schur geometrically convex (Schur geometrically concave) on with respect to if . Chu et al.  pointed out that this conclusion is also true for . Shi et al. [5, 16], Li and Shi , and Gu and Shi  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 ). (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 . 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 ) such as the following lemma.

Lemma 3.2 (see ). 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 , 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 , 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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