1. Introduction

Throughout this paper, let be a non-empty open interval in . A class of functions are defined as

(1.1)

where

If or , in ^[1] chu ed al. proved

Theorem A([1, Theorem 1.1]). Suppose that is an open interval and is a continuous function. The function

(1.2)

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

Theorem B ([1, Theorem 1.2.]). Suppose that is an open interval and is a continuous function. The function

(1.3)

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

If , in ^[2] I. Franjić and J. Pečarić proved

Theorem C ([2, Theorem 4.]). If , then the following statements are equivalent:

(a) The function

(1.4)

is Schur-convex on .

(b) For all , we have

(c) The function is 4-convex on .

Remark ([2, Remark3.]). In ^[2], I. Franjić and J. Pečarić have proven that being a convex function is not a sufficient condition for to be Schur-convex.

Schur-convexity has aroused the interest of many researchers, and numerous papers have been devoted to it. For example, in ^{[3, 4, 6, 7]}, some related results were given.

The purpose of this paper is to prove the following results:

Theorem 1.1. Let be an open interval, and be a twice differentiable mapping such that is integrable. If and is convex (concave) on , then function is Schur-convex (concave) on .

Theorem 1.2. Let be an open interval, and be a twice differentiable mapping such that is integrable. If , and the function is Schur-convex (concave) on , then is convex (concave) on .

2. Definitions and Lemmas

In order to prove our result, we shall need several Definitions and Lemmas, which we present in this section.

Definition 2.1(^[5]). Let . is said to be majorized by (in symbols ) if

and

where and are rearrangements of and in a descending order.

Definition 2.2(^[5]). Let . The function is said to be a Schur-convex function on if on implies . is said to be a Schur-concave function on if and only if is Schur-convex.

Lemma 2.1 (Schur-Ostrowski Theorem) (^[5]). Let be a symmetric convex set with nonempty interior, and be a continuous symmetric function on . If is differentiable in . Then is Schur-convex (concave) on if and only if

for all .

Lemma 2.2. Let be an open interval, with . If and is a twice differentiable mapping such that is integrable, then the following identity holds

(2.1)

where

Proof. It suffices to note that

(2.2)

From integrating by part the right-hand sides of (2.2), we can state:

(2.3)

and

(2.4)

Using (2.3) and (2.4) in (2.2), it follows that

Thus, using the change of the variable for and by multiplying both sides by , we have the conclusion (2.1).

This completes the proof.

Lemma 2.3. Let be an open interval, and be a twice differentiable mapping such that is integrable. If , and with . Then it follows that

where .

Proof. From and , then

(2.5)

and

(2.6)

So, by Integral mean value theorem, we have

where .

This completes the proof.

Lemma 2.4. Suppose that be defined as in (1.1). If has continuous second order derivatives on , then

for all .

Proof. For any , from (1) together with the L’Hospital’s rule we clearly see that

Making use of similar arguments for , and we get

This completes the proof.

3. Proof of Theorems

Proof of Theorem 1.1. The proof is divided into three cases.

Case 1. If , then Lemma 2.4 leads to

Case 2. If , then (1.1) leads to

(3.1)

and

(3.2)

By Lemma 2.1 and Lemma 2.2, we have

If andis convex, then

Therefore, is Schur convex on from Lemma 2.1.

Case 3. If , since is a symmetric function, we also can conclude that is Schur convex on from and the convexity of .

It follows from the similar arguments as above that is Schur-concave on if and is concave on , which completes the proof.

Proof of Theorem 1.2. If is Schur-convex on , then from (1.1) and Definition 1 together with the fact that we have

If by Lemma 2.3, we have

where .

Namely , and so, sinceandare arbitrary, we conclude that is convex.

Since is a symmetric function, if we also can conclude that is convex.

It follows from the similar arguments as above that is concave on if is Schur- concave on , which completes the proof.

4. Conclusion

In this paper, we discuss Schur convexity for a class of symmetric functions and obtain the following results:

Theorem 1.1. Let be an open interval, and be a twice differentiable mapping such that is integrable. If and is convex (concave) on , then function is Schur-convex (concave) on .

Theorem 1.2. Let be an open interval, and be a twice differentiable mapping such that is integrable. If , and the function is Schur-convex (concave) on , then is convex (concave) on .

Acknowledgement

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, China.

Competing Interests

The authors declare that they have no competing interests.

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