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: