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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 108920, 12 pages
http://dx.doi.org/10.1155/2010/108920
Research Article

Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 29 March 2010; Revised 23 July 2010; Accepted 29 July 2010

Academic Editor: Lance Littlejohn

Copyright © 2010 Yu-Ming Chu 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 answer the question: for , what are the greatest value and the least value such that the double inequality holds for all with . Here, , , and denote the power of order , Seiffert, and geometric means of two positive numbers and , respectively.

1. Introduction

For , the power mean of order and the Seiffert mean of two positive numbers and are defined by The main properties of the power mean are given in [1]. It is well known that is strictly increasing with respect to for fixed with . Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for the power mean can be found in the literature [216].

The Seiffert mean was introduced by Seiffert in [17], it can be rewritten in the following symmetric form (see [18, ]):

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

In [9], Alzer and Janous presented the sharp power mean bounds for the sum as follows: for all with .

In [17], Seiffert proved that for all with .

The following power mean bounds for the Seiffert mean was given by Jagers [19]: for all with .

In [20, 21], the authors presented the bounds for the Seiffert mean in terms of and as follows: for all with .

The following sharp lower power mean bounds for , , and can be found in [4, 6]: for all with .

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

2. Lemmas

In order to prove our main result, we need several lemmas which we present in this section.

Lemma 2.1. Let , and . Then there exists such that for , for and .

Proof. Simple computations lead to
Inequality (2.5) implies that is strictly increasing in . Then (2.3) and (2.4) lead to that there exists such that for and for . Hence, is strictly decreasing in and strictly increasing in .
Therefore, Lemma 2.1 follows from (2.1) and (2.2) together with the monotonicity of .

Lemma 2.2. If , then the following statements are true:(1);(2);(3).

Proof. Simple computations lead to + = ; + + = ; + + = .

Lemma 2.3. If , then holds for all with .

Proof. Without loss of generality, we assume that . Let and . Then Let Then simple computations lead to where  /  + , where + + , Let and . Then
From (2.21)–(2.23) we clearly see that for , hence is strictly increasing in . Then (2.20) implies that for , hence is strictly increasing in .
It follows from (2.18) and the monotonicity of that for , hence is strictly increasing in . Then (2.17) implies that for , therefore is strictly increasing in .
Equation (2.16) and the monotonicity of lead to that for , so is strictly increasing in .
From (2.9)–(2.14) and the monotonicity of we can deduce that for .
Therefore, Lemma 2.3 follows from (2.7) and (2.8) together with (2.24).

Remark 2.4. In [22, Theorem], the authors proved that for all with .
Obviously, (2.6) is the generalization of (2.25)–(2.27).

Remark 2.5. If , then and (2.17) leads to
Inequality (2.28) and the continuity of imply that there exists such that for .
From (2.29) and (2.9)–(2.16) we can deduce that for .
Equations (2.7) and (2.8) together with (2.30) lead to for all with .
Therefore, is the largest value in such that inequality (2.6) holds for .

3. Main Result

Theorem 3.1. If , then holds for all with , and and are the best possible lower and upper power mean bounds for the product .

Proof. For all with , from (1.3), (1.5) and Lemma 2.1 we clearly see that for all , and for .
Next, we prove that (3.3) is also true for and all with .
Without loss of generality, we assume that . Let and . Then (1.1) leads to Let Then simple computations lead to From Lemma 2.1 we know that there exists such that for , and for .
We divide two cases to prove that for .

Case 1. . Then (3.11) follows from (3.7) and (3.8) together with (3.10).

Case 2. . Then (3.7) can be written as where Let , then (3.13) leads to where Let , then (3.15) leads to
From and , we clearly see that , , and . These inequalities and Lemma 2.2 lead to From (3.16)–(3.18) we can deduce that for .
Equation (3.14) together with (3.19) imply that for .
Therefore, (3.11) follows from (3.9) and (3.12) together with (3.20).
It follows from (3.4)–(3.6) and (3.11) that for and all with .
At last, we prove that and are the best possible lower- and upper-power mean bounds for the product , respectively.
For any and , from (1.1) one has where  (1/2) .
Let , making use of the Taylor expansion we get
Equation (3.22) implies that for any there exists such that for .
Equations (3.23) and (3.24) imply that for any there exists such that for .

Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions. This paper is partly supported by N S Foundation of China under Grant 60850005, N S Foundation of Zhejiang Province under Grants Y7080106 and Y607128, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

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