Schur convexity of the generalized geometric Bonferroni mean and the relevant inequalities.

In this paper, we discuss the Schur convexity, Schur geometric convexity and Schur harmonic convexity of the generalized geometric Bonferroni mean. Some inequalities related to the generalized geometric Bonferroni mean are established to illustrate the applications of the obtained results.

Introduction

The Schur convexity of functions relating to special means is a very significant research subject and has attracted the interest of many mathematicians. There are numerous articles written on this topic in recent years; see [1, 2] and the references therein. As supplements to the Schur convexity of functions, the Schur geometrically convex functions and Schur harmonically convex functions were investigated by Zhang and Yang [3], Chu, Zhang and Wang [4], Chu and Xia [5], Chu, Wang and Zhang [6], Shi and Zhang [7, 8], Meng, Chu and Tang [9], Zheng, Zhang and Zhang [10]. These properties of functions have been found to be useful in discovering and proving the inequalities for special means (see [11-14]).

Recently, it has come to our attention that a type of means which is symmetrical on n variables and involves two parameters, it was initially proposed by Bonferroni [15], as follows: where , , , and .

is called the Bonferroni mean. It has important application in multi criteria decision-making (see [16-21]).

Beliakov, James and Mordelová et al. [22] generalized the Bonferroni mean by introducing three parameters p, q, r, i.e., where , , , and .

Motivated by the Bonferroni mean and the geometric mean , Xia, Xu and Zhu [23] introduced a new mean which is called the geometric Bonferroni mean, as follows: where , , , and .

An extension of the geometric Bonferroni mean was given by Park and Kim in [19], which is called the generalized geometric Bonferroni mean, i.e., where , , , and .

Remark 1

For , it is easy to observe that

Remark 2

If , , then the generalized geometric Bonferroni mean reduces to the geometric mean, i.e.,

Remark 3

If , then

For convenience, throughout the paper denotes the set of real numbers, denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as

In a recent paper [24], Shi and Wu investigated the Schur m-power convexity of the geometric Bonferroni mean . The definition of Schur m-power convex function is as follows:

Let be a function defined by Then a function is said to be Schur m-power convex on Ω if for all and implies .

If −φ is Schur m-power convex, then we say that φ is Schur m-power concave.

Shi and Wu [24] obtained the following result.

Proposition 1

For fixed positive real numbers p, q, (i) if or , then is Schur m-power convex on ; (ii) if or , then is Schur m-power concave on .

In this paper we discuss the Schur convexity, Schur geometric convexity and Schur harmonic convexity of the generalized geometric Bonferroni mean . Our main results are as follows.

Theorem 1

For fixed non-negative real numbers p, q, r with , if , , then is Schur concave, Schur geometric convex and Schur harmonic convex on .

Corollary 1

For fixed non-negative real numbers p, q with , if , , then is Schur concave, Schur geometric convex and Schur harmonic convex on .

Preliminaries

We introduce some definitions, lemmas and propositions, which will be used in the proofs of the main results in subsequent sections.

Definition 1

(see [1])

Let and .

is said to be majorized by (in symbols ) if for and , where and are rearrangements of and in a descending order.

Let ,the function φ: is said to be Schur convex on Ω if on Ω implies . φ is said to be Schur concave function on Ω if and only if −φ is Schur convex function on Ω.

Definition 2

(see [1])

Let and . is said to be a convex set if and imply

Definition 3

(see [1])

(i) A set is called symmetric, if implies for every permutation matrix P.

(ii) A function is called symmetric if for every permutation matrix P and for all .

The following proposition is called Schur’s condition. It provides an approach for testing whether a vector valued function is Schur convex or not.

Proposition 2

(see [1])

Let be symmetric and have a nonempty interior convex set. is the interior of Ω. is continuous on Ω and differentiable in . Then φ is the Schur convex function (Schur concave function) if and only if φ is symmetric on Ω and holds for any .

Definition 4

(see [25])

Let and .

is called a geometrically convex set if for all , and α, such that .

Let . The function φ: is said to be Schur geometrically convex function on Ω if on Ω implies . The function φ is said to be a Schur geometrically concave function on Ω if and only if −φ is Schur geometrically convex function.

Proposition 3

(see [25])

Let be a symmetric and geometrically convex set with a nonempty interior . Let be continuous on Ω and differentiable in . If φ is symmetric on Ω and holds for any , then φ is a Schur geometrically convex (Schur geometrically concave) function.

Definition 5

(see [26])

Let .

A set Ω is said to be harmonically convex if for every and , where and

A function is said to be Schur harmonically convex on Ω if implies . A function φ is said to be a Schur harmonically concave function on Ω if and only if −φ is a Schur harmonically convex function.

Proposition 4

(see [26])

Let be a symmetric and harmonically convex set with inner points, and let be a continuously symmetric function which is differentiable on . Then φ is Schur harmonically convex (Schur harmonically concave) on Ω if and only if holds for any .

Remark 4

Propositions 3 and 4 provide analogous Schur’s conditions for determining Schur geometrically convex functions and Schur harmonically convex functions, respectively.

Lemma 1

(see [1])

Let and . Then

Lemma 2

(see [1])

If , , then, for any non-negative constant c satisfying , one has

Proof of main result

Proof of Theorem 1

Note that the generalized geometric Bonferroni mean is defined by taking the natural logarithm gives where

Differentiating with respect to and , respectively, we have

It is easy to see that is symmetric on . For , we have

This implies that for (). By Proposition 2, we conclude that is Schur concave on .

In view of the discrimination criterion of Schur geometrically convexity, we start with the following calculations:

Thus, we have for (). It follows from Proposition 3 that is Schur geometric convex on .

Finally, we discuss the Schur harmonic convexity of . A direct computation gives

Hence, we obtain for (). Using Proposition 4 leads to the assertion that is Schur harmonic convex on .

The proof of Theorem 1 is completed. □

Remark 5

As a direct consequence of Theorem 1, taking in Theorem 1 together with the identity , we arrive at the assertion of Corollary 1.

Applications

As an application of Theorem 1, we establish the following interesting inequalities for generalized geometric Bonferroni mean.

Theorem 2

Let p, q, r be non-negative real numbers with . Then, for arbitrary (),

Proof

It follows from Theorem 1 that is Schur concave on .

Using Lemma 1, one has

Thus, we deduce from Definition 1 that which implies that

Theorem 2 is proved. □

Theorem 3

Let p, q, r be non-negative real numbers with , and let c be a constant satisfying , . Then, for arbitrary (),

By the majorization relationship given in Lemma 2, it follows from Theorem 1 that that is, which implies that

This completes the proof of Theorem 3. □

Conclusion

This paper is a follow-up study of our recent work [24], we generalize the geometric Bonferroni mean by introducing three non-negative parameters p, q, r, under the condition of , we prove that the generalized geometric Bonferroni mean is Schur concave, Schur geometric convex and Schur harmonic convex on . As an application of the Schur convexity, we establish two inequalities for generalized geometric Bonferroni mean. In fact, there have been a large number inequalities for means which originate from the Schur convexity of functions. For details, we refer the interested reader to [27-32] and the references therein.