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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 438971, 3 pages
http://dx.doi.org/10.1155/2013/438971
Research Article

Optimal Two Parameter Bounds for the Seiffert Mean

School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China

Received 30 April 2013; Revised 18 July 2013; Accepted 18 July 2013

Academic Editor: Zhihua Zhang

Copyright © 2013 Hui Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain sharp bounds for the Seiffert mean in terms of a two parameter family of means. Our results generalize and extend the recent bounds presented in the Journal of Inequalities and Applications (2012) and Abstract and Applied Analysis (2012).

1. Introduction

For with , the Seiffert mean , root mean square , and contraharmonic mean are defined by respectively. It is well known that the inequalities hold for all with .

Recently, , , and have been the subject of intensive research. In particular, many remarkable inequalities and properties for these means can be found in the literature [18].

For , very recently Chu et al. [9, 10] proved that the inequalities hold for all with if and only if , , , and .

Let , , and where is the classical arithmetic mean of and . Then from (2), (3), and (7) we clearly see that and is strictly increasing with respect to for fixed with .

It is natural to ask what are the greatest value and the least value in such that the double inequality holds for all with and . The aim of this paper is to answer this question; our main result is the following Theorem 1.

Theorem 1. If and , then the double inequality holds for all with if and only if and .

Remark 2. If we take and in Theorem 1, then inequality (10) reduces to inequalities (5) and (6), respectively.

2. Proof of Theorem 1

In order to prove Theorem 1 we need two lemmas, which we present in this section.

Lemma 3 (see [11, Theorem  1.25]). For , let be continuous on and differentiable on ; let on . If is increasing (decreasing) on , then so are If is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 4. Let , and Then(1) for if and only if ;(2) for if and only if .

Proof. By (12) and simple computations one has where
Let and . Then is strictly increasing in :
It is not difficult to verify that the function is strictly increasing from onto ; hence (18) implies that is strictly decreasing in . Therefore, is strictly decreasing in follows from Lemma 3 and (17) together with the monotonicity of and . Moreover, making use of l'Hôpital's rule we get
We divide the proof into three cases.
Case  1 (). Then from (15) and (19) together with the monotonicity of we conclude that is strictly increasing in . Therefore for follows from (13) and the monotonicity of .
Case  2 (). Then from (15) and (20) together with the monotonicity of we clearly see that is strictly decreasing in . Therefore for follows from (13) and the monotonicity of .
Case  3 (). Then from (15), (19), and (20) together with the monotonicity of we know that there exists such that is strictly decreasing in and strictly increasing in . Therefore, in if and only if follows from (13) and the piecewise monotonicity of , which (14) gives immediately .

Proof of Theorem 1. Since both and are symmetric and homogeneous of degree 1, without loss of generality, we assume that . Let . Then from (1) and (7) we get
Therefore, Theorem 1 follows from Lemma 4 and (21).

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant nos. 11071069 and 11171307) and the Natural Science Foundation of Zhejiang Province (Grant no. LY13A010004).

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