Research Article | Open Access

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

**Academic Editor:**Lance Littlejohn

#### 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 [2–16].

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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#### Copyright

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.