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Journal of Applied Mathematics
Volume 2012, Article ID 917120, 7 pages
http://dx.doi.org/10.1155/2012/917120
Research Article

Optimal Bounds for Seiffert Mean in terms of One-Parameter Means

1Acquisitions & Cataloging Department of Library, Huzhou Teachers College, Huzhou 313000, China
2Department of Basic Course Teaching, Tongji Zhejiang College, Jiaxing 314000, China
3Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 11 July 2012; Accepted 7 September 2012

Academic Editor: Francisco J. Marcellán

Copyright © 2012 Hua-Nan Hu 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

The authors present the greatest value and the least value such that the double inequality holds for all with , where and denote the Seiffert and pth one-parameter means of two positive numbers a and b, respectively.

1. Introduction

For the th one-parameter mean and the Seiffert mean of two positive real numbers and are defined by respectively. Recently, both mean values have been the subject of intensive research. In particular, many remarkable inequalities and properties for and can be found in the literature [114].

It is well known that the one-parameter mean is continuous and strictly increasing with respect to for fixed , with . Many mean values are the special case of the one-parameter mean, for example:

Seiffert [4] proved that the double inequality holds for all with , where and is the th power mean of and .

In [1517], the authors presented the best possible bounds for the Seiffert mean in terms of the Lehmer, power-type Heron, and one-parameter Gini means as follows: for all with , where , and , and and denote the Lehmer, power-type Heron, and one-parameter Gini means of and , respectively.

The purpose of this paper is to answer the question: what are the greatest value and the least value such that the double inequality holds for all with ?

2. Lemma

In order to establish our main result we need the following lemma.

Lemma 2.1. If , and + + + , then there exists such that for and for .

Proof. Let , and . Then simple computations lead to for .
From the inequality (2.21) we clearly see that is strictly decreasing in . Then (2.19) and (2.20) lead to the conclusion that there exists such that for and for . Hence, is strictly increasing in and strictly decreasing in .
It follows from (2.16) and (2.17) together with the monotonicity of that there exists such that for and for . Therefore, is strictly increasing in and strictly decreasing in .
From (2.13) and (2.14) together with the monotonicity of we know that there exists such that for and for . So, is strictly increasing in and strictly decreasing in .
Equations (2.10) and (2.11) together with the monotonicity of imply that there exists such that for and for . Hence, is strictly increasing in and strictly decreasing in .
It follows from (2.7) and (2.8) together with the monotonicity of that there exists such that for and for . Therefore, is strictly increasing in and strictly decreasing in .
From (2.4) and (2.5) together with the monotonicity of we clearly see that there exists such that for and for . So, is strictly increasing in and strictly decreasing in .
Therefore, Lemma 2.1 follows from (2.1) and (2.2) together with the monotonicity of .

3. Main Result

Theorem 3.1. The double inequality holds for all with , and and are the best possible lower and upper one-parameter mean bounds for the Seiffert mean , respectively.

Proof. Without loss of generality, we assume that . Let . Then from (1.1) and (1.2) we have Let Then simple computations lead to for .
Therefore, for all with follows from (3.2)–(3.4).
Next, we prove that for all with .
Let . Then (1.1) and (1.2) lead to Let Then simple computations lead to where is defined as in Lemma 2.1.
From Lemma 2.1 and (3.9) we know that there exists such that is strictly increasing in and strictly decreasing in . Then (3.8) leads to that for .
Therefore, the inequality (3.5) follows from (3.6), (3.7), and (3.10).
Finally, we prove that and are the best possible lower and upper one-parameter mean bounds for the Seiffert mean , respectively.
Let , and . Then from (1.1) and (1.2) one has where
Letting and making use of Taylor expansion we get
The inequality (3.11) implies that for any , there exists , such that for .
Equations (3.12)–(3.14) imply that for any , there exists such that for .

Acknowledgments

This paper was supported by the Natural Science Foundation of Huzhou Teachers College (Grant no. 2010065) and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant no. Y201016277).

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