/ / Article

Research Article | Open Access

Volume 2010 |Article ID 108920 | https://doi.org/10.1155/2010/108920

Yu-Ming Chu, Ye-Fang Qiu, Miao-Kun Wang, "Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means", Abstract and Applied Analysis, vol. 2010, Article ID 108920, 12 pages, 2010. https://doi.org/10.1155/2010/108920

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

Revised23 Jul 2010
Accepted29 Jul 2010
Published20 Sep 2010

#### 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 . 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 .

The Seiffert mean was introduced by Seiffert in , 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 , Alzer and Janous presented the sharp power mean bounds for the sum as follows: for all with .

In , Seiffert proved that for all with .

The following power mean bounds for the Seiffert mean was given by Jagers : 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 .

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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