Keywords: Schurconvex function, Schurgeometrically convex function, Schurharmonically convex function, inequality, generalized logarithmic mean
Received July 12, 2015; Revised August 15, 2015; Accepted August 27, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
1. Introduction
In ^{[3]}, N. Elezović and J. Pečarić established the following theorem.
Theorem A (^{[3]}). Let and . Then
is Schurconvex (Schurconcave) 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 Schurconvex (Schurconcave) on if and only if
holds (reverses) for all .
For more information on this topic, please refer to ^{[5, 8, 9]} and closelyrelated references therein.
In this paper, we discuss Schurgeometric and Schurharmonic 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 Schurconvex function on if on implies . is said to be a Schurconcave function on if and only is Schurconvex function.
Definition 2 (^{[1, 2]}). Let, and and let .
(1) is said to be a Schurgeometrically convex function on if on implies . is said to be a Schurgeometrically concave function on if and only is Schurgeometrically convex function.
(2) is said to be a Schurharmonically convex function on if on implies . is said to be a Schurharmonically concave function on if and only is Schurharmonically convex function.
Lemma 2.1(^{[1]}). Let be a continuous function on and differentiable in interior of . Then is Schurgeometrically convex (Schurgeometrically 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 Schurharmonically convex (Schurharmonically 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 Schurgeometrically convex and Schurharmonically convex on .
3. Main Results
Theorem 3.1. Let and be defined in Theorem A.
(i). If is convex and increasing on, then is Schurgeometrically convex on .
(ii). Ifis concave and decreasing on, then is Schurgeometrically 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 Schurgeometrically convex on . If is concave and decreasing on , then the inequality (4) is reversed. According to Lemma 2.1, it follows that is Schurgeometrically 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 Schurharmonically convex on .
(ii). If is concave and decreasing on , then is Schurharmonically concave on .
Proof . If and , we have For all , if is convex and increasing, using inequality (3) and (2), we get
 (5) 
Therefore, is Schurharmonically 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 Schurharmonically concave function on. The proof of Theorem 3.2 is complete.
4. Applications
Theorem 4.1. For and , if , then is Schurgeometrically convex and Schurharmonically 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 Schurgeometrically convex and Schurharmonically convex on for , then is Schurgeometrically convex and Schurharmonically 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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