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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 425175, 6 pages
http://dx.doi.org/10.1155/2012/425175
Research Article

Sharp Bounds for Seiffert Mean in Terms of Contraharmonic Mean

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2Department of Mathematics, Hangzhou Normal University, Hangzhou 310012, China

Received 17 September 2011; Accepted 5 November 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Yu-Ming Chu and Shou-Wei Hou. 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 find the greatest value and the least value in such that the double inequality holds for all with . Here, 2 arctan and are the Seiffert and contraharmonic means of and , respectively.

1. Introduction

For with , the Seiffert mean and contraharmonic mean are defined by

respectively. Recently, both mean values have been the subject of intensive research. In particular, many remarkable inequalities and properties for these means can be found in the literature [112].

Let , , , and let and be the arithmetic, geometric, square root, and th power means of two positive numbers and , respectively. Then it is well known that is continuous and strictly increasing with respect to for fixed with , and the inequalities hold for all with .

Seiffert [12] proved that the double inequality holds for all with .

Hästö [13] proved that the function is increasing in if .

In [14], the authors found the greatest value and the least value such that the double inequality holds for all with . Here, , and is the th power-type Heron mean of and .

Wang et al. [15] answered the question: what are the best possible parameters and such that the double inequality holds for all with , where is the th Lehmer mean of and .

In [16, 17], the authors proved that the inequalities hold for all with if and only if , , , , and .

For fixed with , let and

Then it is not difficult to verify that is continuous and strictly increasing in . Note that and . Therefore, it is natural to ask what are the greatest value and the least value in such that the double inequality holds for all with . The main purpose of this paper is to answer this question. Our main result is the following Theorem 1.1.

Theorem 1.1. If , then the double inequality holds for all with if and only if and .

2. Proof of Theorem 1.1

Proof of Theorem 1.1. Let and . We first proof that the inequalities hold for all with .
From (1.1) and (1.2) we clearly see that both and are symmetric and homogenous of degree 1. Without loss of generality, we assume that . Let and , then from (1.1) and (1.2) one has Let Then simple computations lead to where
Let , , . Then from (2.8) we get
We divide the proof into two cases.
Case 1 (). Then (2.6), (2.13), and (2.15) lead to
Note that
It follows from (2.8), (2.10), (2.12), (2.14), and (2.19) that
From (2.14) and inequality (2.19), we clearly see that is strictly increasing in . Then (2.18) and (2.23) lead to the conclusion that there exists such that for and for . Hence, is strictly decreasing in and strictly increasing in .
It follows from (2.17) and (2.22) together with the piecewise monotonicity of that there exists such that is strictly decreasing in and strictly increasing in .
From (2.11) and (2.21) together with the piecewise monotonicity of , we conclude that there exists such that is strictly decreasing in and strictly increasing in .
Equations (2.7), (2.9), and (2.20) together with the piecewise monotonicity of imply that there exists such that is strictly decreasing in and strictly increasing in .
Therefore, inequality (2.1) follows from (2.3)–(2.5) and (2.16) together with the piecewise monotonicity of .
Case 2 (). Then (2.8) leads to for .
Inequality (2.24) and (2.7) imply that is strictly increasing in . Therefore, inequality (2.2) follows from (2.3)–(2.5) together with the monotonicity of .
From inequalities (2.1) and (2.2) together with the monotonicity of in , we know that inequality (1.8) holds for all , , and all with .
Next, we prove that is the best possible parameter in such that inequality (2.1) holds for all with .
For any , from (2.6) one has
Equations (2.3) and (2.4) together with inequality (2.25) imply that for any there exists such that for .
Finally, we prove that is the best possible parameter such that inequality (2.2) holds for all with .
For any , from (2.13) one has
From inequality (2.27) and the continuity of , we know that there exists such that for .
Equations (2.3)–(2.5), (2.7), (2.9), and (2.11) together with inequality (2.28) imply that for any there exists such that for .

Acknowledgments

This research was supported by the National Natural Science Foundation of China under Grants 11071069 and 11171307, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

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