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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 182905, 8 pages
http://dx.doi.org/10.1155/2012/182905
Research Article

Optimal Inequalities for Power Means

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2School of Mathematics Science, Anhui University, Hefei 230039, China
3Department of Mathematics, Hunan City University, Yiyang 413000, China

Received 9 December 2011; Revised 30 January 2012; Accepted 2 February 2012

Academic Editor: Hector Pomares

Copyright © 2012 Yong-Min Li 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 present the best possible power mean bounds for the product for any , , and all with . Here, is the th power mean of two positive numbers and .

1. Introduction

For , the th power mean of two positive numbers and is defined by

It is well known that is continuous and strictly increasing with respect to for fixed with . Many classical means are special cases of the power mean, for example, , and are the harmonic, geometric and arithmetic means of and , respectively. Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities and properties for the power mean can be found in literature [122].

Let , and be the logarithmic, Seiffert and identric means of two positive numbers and with , respectively. Then it is well known that for all with .

In [2329], the authors presented the sharp power mean bounds for , , and as follows: for all with .

Alzer and Qiu [12] proved that the inequality holds for all with if and only if .

The following sharp bounds for the sum , and the products and in terms of power means were proved in [5, 8]: for any and all with .

In [2, 7] the authors answered the questions: for any , what are the greatest values , , , and , and the least values , , , and , such that the inequalities hold for all with ?

It is the aim of this paper to present the best possible power mean bounds for the product for any , and all with .

2. Main Result

Theorem 2.1. Let , and with . Then(1) for ,(2) for and for , and the bounds and for the product in either case are best possible.

Proof. From (1.1) we clearly see that is symmetric and homogenous of degree 1. Without loss of generality, we assume that , .(1)If , then (1.1) leads to (2)Firstly, we compare the value of to the value of for . From (1.1) we have
Let then simple computations lead to where where
If , then (2.9) implies that is strictly decreasing in . Therefore, follows easily from (2.2)–(2.8) and the monotonicity of .
If , then (2.9) leads to the conclusion that is strictly increasing in . Therefore, follows easily from (2.2)–(2.8) and the monotonicity of .
Secondly, we compare the value of to the value of . It follows from (1.1) that
Let then simple computations lead to
If , then (2.13) implies that is strictly increasing in . Therefore, follows easily from (2.10)–(2.12) and the monotonicity of .
If , then (2.13) leads to the conclusion that is strictly decreasing in . Therefore, follows easily from (2.10)–(2.12) and the monotonicity of .
Next, we prove that the bound for the product in either case is best possible.
If , then for any and we have
Letting and making use of Taylor’s expansion, one has
Equations (2.14) and (2.15) imply that for any and there exists , such that for .
If , then for any and we have
Letting and making use of Taylor’s expansion, one has
Equations (2.16) and (2.17) imply that for any and there exists , such that for .
Finally, we prove that the bound for the product in either case is best possible.
If , then for any we clearly see that
Equation (2.18) implies that for any and there exists , such that for .
If , then for any we have
Equation (2.19) implies that for any and there exists , such that for .

Acknowledgments

This paper was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, the Natural Science Foundation of Hunan Province under Grant 09JJ6003, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

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