1. Introduction

In ^[3], N. Elezović and J. Pečarić established the following theorem.

Theorem A (^[3]). Let and . Then

is Schur-convex (Schur-concave) on if and only if is convex (concave) on

In ^{[7, 10]}, Theorem A was generalized as the following theorem.

Theorem B (^{[7, 10]}). Let be a continuous function and a positive continuous weight on an interval . Then the weighted arithmetic mean of with weight defined by

is Schur-convex (Schur-concave) on if and only if

holds (reverses) for all .

For more information on this topic, please refer to ^{[5, 8, 9]} and closely-related references therein.

In this paper, we discuss Schur-geometric and Schur-harmonic convexity of the mean and obtain two results which generate Theorem A.

2. Definitions and Lemmas

In order to prove our main results we need the following definitions and lemmas.

Definition 1 (^[4]). Let and , and let

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

(2) means for all is said to be increasing if implies . is said to be decreasing if and only is increasing.

(3) is said to be a Schur-convex function on if on implies . is said to be a Schur-concave function on if and only is Schur-convex function.

Definition 2 (^{[1, 2]}). Let, and and let .

(1) is said to be a Schur-geometrically convex function on if on implies . is said to be a Schur-geometrically concave function on if and only is Schur-geometrically convex function.

(2) is said to be a Schur-harmonically convex function on if on implies . is said to be a Schur-harmonically concave function on if and only is Schur-harmonically convex function.

Lemma 2.1(^[1]). Let be a continuous function on and differentiable in interior of . Then is Schur-geometrically convex (Schur-geometrically concave) if and only if it is symmetric and

for all

Lemma 2.2 (^[2]). Let be a continuous function on and differentiable in interior of . Then is Schur-harmonically convex (Schur-harmonically concave) if and only if it is symmetric and

for all

For two positive numbers and , define

and

It is well known that ,,and are respectively called the arithmetic, geometric, harmonic and generalized logarithmic means of and

Lemma 2.3 (^[6]) is increasing function on .

In this paper, we will prove that the function is Schur-geometrically convex and Schur-harmonically convex on .

3. Main Results

Theorem 3.1. Let and be defined in Theorem A.

(i). If is convex and increasing on, then is Schur-geometrically convex on .

(ii). Ifis concave and decreasing on, then is Schur-geometrically concave on.

Proof. If and , we have For all , a straightforward computation gives

(3)

If is convex and increasing on , by the inequality (2), we obtain

(4)

Hence, is Schur-geometrically convex on . If is concave and decreasing on , then the inequality (4) is reversed. According to Lemma 2.1, it follows that is Schur-geometrically concave . This completes the proof of Theorem 3.1.

Theorem 3.2. Let and be defined in Theorem A.

(i). If is convex and increasing on , then is Schur-harmonically convex on .

(ii). If is concave and decreasing on , then is Schur-harmonically concave on .

Proof . If and , we have For all , if is convex and increasing, using inequality (3) and (2), we get

(5)

Therefore, is Schur-harmonically convex function on . If is concave and decreasing function on, then the inequality (5) is reversed. According to Lemma 2.2, it follows that is Schur-harmonically concave function on. The proof of Theorem 3.2 is complete.

4. Applications

Theorem 4.1. For and , if , then is Schur-geometrically convex and Schur-harmonically convex.

Proof. Taking for all , if , it follows that

and is convex increasing on for . Therefore, by Theorem 3.1 and 3.2, we have

is Schur-geometrically convex and Schur-harmonically convex on for , then is Schur-geometrically convex and Schur-harmonically convex on for . Thus, Theorem 4.1 is proved.

Corollary . Forand, define , , , , ， and for . Then

(1) when and , we have

(2) when , we have

Proof. When , it is easy to obtain that When and , by Corollary ^[6] and Lemma 2.3, Corollary is thus proved.

Acknowledgements

The authors thank the anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

Support

This work was partially supported by the National Natural Science Foundation of China under Grant No. 11361038 and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123 and No. 2014BS0106, China.

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