The Schur-Convexity of the Generalized Muirhead-Heronian Means

Abstract

We give a unified generalization of the generalized Muirhead means and the generalized Heronian means involving three parameters. The Schur-convexity of the generalized Muirhead-Heronian means is investigated. Our main result implies the sufficient conditions of the Schur-convexity of the generalized Heronian means and the generalized Muirhead means.

1. Introduction

In what follows, we denote the set of real numbers by , the set of nonnegative real numbers by , and the set of positive real numbers by .

Let ; the classical Heronian means is defined by (see [1])

In 1999, Mao [2] gave the definition of dual Heronian means; that is,

In 2001, Janous [3] considered the unified generalization of Heronian means and and presented a weighted generalization of the above-mentioned Heronian-type means, as follows:

Jia and Cao [4] investigated the exponential generalization of Heronian means and they established some related inequalities. The monotonicity and Schur-convexity of the Heronian means were discussed by Li et al. in [5].

Shi et al. [6] discussed the Schur-convexity of a further generalization of the Heronian means given by and they obtained a significant result asserted by Theorem A below.

Theorem A. For fixed ,(1)if , then is Schur-convex for .(2)If , then is Schur-concave for .

As a further investigation of Theorem A, Fu et al. [7] gave the necessary and sufficient condition for the Schur-convexity of the generalized Heronian means , which is stated in the following theorem.

Theorem B. For fixed , the generalized Heronian means is Schur-convex for if and only if
Furthermore, is Schur-concave for if and only if

Remark 1. It is easy to observe that, for , , is Schur-convex and Schur-concave for . In addition, we note that . Thus, the conditions of Schur-convexity of in Theorem B can be rewritten as
The Schur-power-convexity of was investigated by Yang [8].
In 2006, Trif [9] considered the following generalized Muirhead means, defined by where , , .
Gong et al. [10] investigated the Schur-convexity of generalized Muirhead means and obtained the following results.

Theorem C. For fixed , the generalized Muirhead means is Schur-convex for if and only if
Furthermore, is Schur-concave for if and only if

Remark 2. If we define, for , , the generalized Muirhead means by , we can easily find that is Schur-concave for ; thereby, the conditions of Schur-concave of in Theorem C can be rewritten as
The Schur-geometric-convexity and Schur-harmonic-convexity of the generalized Muirhead means were studied by Xia and Chu in [11, 12].
In this paper we generalize the generalized Muirhead means and the generalized Heronian means in a unified form. For this purpose we define a generalized Muirhead-Heronian means , as follows: where , .

The paper is organized as follows. Section 2 introduces several definitions and lemmas; Section 3 discusses the Schur-convexity of the generalized Muirhead-Heronian means; Section 4 provides some remarks on the results given in Theorem 9 and it is shown that the sufficient conditions of Schur-convexity of the generalized Heronian means and the generalized Muirhead means can be deduced from Theorem 9 as special cases.

2. Definitions and Lemmas

We introduce and establish several definitions and lemmas, which will be used in the proofs of the main results in Section 3.

Definition 3 (see [13, page 7]). For any , , let and denote the components of and in decreasing order, respectively.
The -tuple is said to majorize (or is to be majorized by ), in symbols , if

Definition 4 (see [13, page 54]). For any , (), is said to be a Schur-convex function on if on implies and is said to be a Schur-concave function on if and only if is a Schur-convex function.

Lemma 5 (see [13, page 57]). Let be a symmetric convex set with nonempty interior and a continuous symmetric function on . If is differentiable on , then is the Schur-convex (Schur-concave) function on if and only if holds for all .

Lemma 6. Suppose that , , and Suppose also that
Then for .

Proof. Differentiating with respect to gives
Let , then
Differentiating with respect to yields
In order to prove Lemma 6, we need to consider the two cases below.
Case  1 (). In view of , we have for . Hence is decreasing on , by which, together with we deduce that for . This means that is increasing on . Thus, we have, for ,
This leads to .
Case  2 (). By , we have for . Thus is increasing on , by which, together with we obtain for . It follows that is increasing on . Thus, we have, for ,
This implies that . The proof of Lemma 6 is complete.

Lemma 7. Suppose that , , and Suppose also that
Then for .

Proof. Using the differential expressions obtained in Lemma 6, one has
We divide the proof of Lemma 7 into four cases.
Case  1. If , then
Thus we have, for , Case  2. If , then
Thus we have, for , Case  3. If , then
Thus we have, for , Case  4. If , then
Thus we have, for , This completes the proof of Lemma 7.

Lemma 8. Suppose that , , and Suppose also that
Then for .

Proof. Based on the differential expressions , , , , , obtained in the proof of Lemma 6, in order to prove Lemma 8, we need to consider the two cases below.
Case  1. If , then
Hence, we deduce that there exists such that , satisfying for and for .
Further, we conclude that is increasing on and decreasing on ; thereby, we get for .
From we have this yields we thus have where
Note that implies and we conclude that for .
Hence, from , one has, for ,
Case  2. If , then
Thus, we deduce that there exists such that , satisfying for and for .
It follows that is decreasing on and increasing on , therefore, we obtain
From , we have that is, we thus have where
Note that implies and
which yields that for .
Therefore, from , one has, for ,
The proof of Lemma 8 is completed.

3. Main Result

The main result of this paper is given by Theorem 9 below.

Theorem 9. For fixed , let and let
The following assertions holds true. (1)If , then the generalized Muirhead-Heronian means is Schur-convex for .(2)If , then the generalized Muirhead-Heronian means is Schur-concave for .

Proof. Note that the expression is of symmetry between and and without loss of generality we assume that .
Case  1. If , then . Define
Hence, is Schur-concave for .
Case  2. If , we have the following known results (see Theorem B and Remark 1 in Section 1).
is Schur-convex for if and only if
is Schur-concave for if and only if Case  3. If , then where where ; in addition, the definition of implies that .
Using Lemma 6 gives where
On the other hand, we deduce from the symmetry of with respect to and that where
Now, by using Lemma 5 and combining the result stated in Case , we deduce that is Schur-convex under the conditions below: This proves the validity of the first assertion in Theorem 9.
It is easy to find that
In view of the symmetry of between and , utilizing a positional exchange between and in the above expression gives
Hence, we deduce from Lemma 7 that where
By using the same method as above, we can deduce that where
Therefore, we deduce from Lemma 5 and the results stated in Cases and that is Schur-concave under the following conditions:
This proves the validity of the second assertion in Theorem 9. The proof of Theorem 9 is thus completed.

4. Some Remarks on the Results of Theorem 9

In this section, we provide some remarks on the results given in Theorem 9; we show that the sufficient conditions of Schur-convexity of the generalized Heronian means and the generalized Muirhead means can be deduced from Theorem 9 as special cases.

Remark 10. If we take in Theorem 9, we have . Furthermore, we have which are the sufficient conditions of Schur-convex of the generalized Heronian means asserted by Theorem B.
On the other hand, we note that, for ,
Hence, the set given in Theorem 9 reduces to the following form:
These are the sufficient conditions of Schur-concave of the generalized Heronian means stated by Theorem B.

Remark 11. If we put in Theorem 9, we have . Furthermore, we have which are the sufficient conditions of Schur-convex of the generalized Muirhead means given by Theorem C.
In addition, for , we have
Thus, the set given in Theorem 9 reduces to the following form:
These are the sufficient conditions of Schur-concave of the generalized Muirhead means asserted by Theorem C.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11171307 and 61374086, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, the Natural Science Foundation of Fujian province under Grant 2012J01014, and the Foundation of Scientific Research Project of Fujian Province Education Department under Grants JK2013051 and JK2012049.