- Research Article
- Open Access

# The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

- Yu-Ming Chu
^{1}Email author, - Ye-Fang Qiu
^{2}, - Miao-Kun Wang
^{2}and - Gen-Di Wang
^{1}

**2010**:436457

https://doi.org/10.1155/2010/436457

© Yu-Ming Chu et al. 2010

**Received:**28 December 2009**Accepted:**22 April 2010**Published:**26 May 2010

## Abstract

We find the greatest value and least value such that the double inequality holds for all with . Here , , and denote the arithmetic, harmonic, and Seiffert's means of two positive numbers and , respectively.

## Keywords

- Real Number
- Lower Power
- Simple Computation
- Positive Real Number
- Optimal Convex

## 1. Introduction

for all with .

for all with .

for all with .

for all with .

for all with .

The purpose of this paper is to find the greatest value and the least value such that the double inequality holds for all with .

## 2. Main Result

Theorem 2.1.

The double inequality holds for all with if and only if and .

Proof.

for all with .

We divide the proof into two cases.

Case 1.

for .

Therefore, inequality (2.1) follows from (2.3)–(2.5) and (2.7) together with (2.9).

Case 2.

From (2.24) and (2.25) we clearly see that for , hence is strictly decreasing in . It follows from (2.22) and (2.23) together with the monotonicity of that there exists such that for and for , hence is strictly increasing in and strictly decreasing in .

From (2.19) and (2.20) together with the monotonicity of we know that there exists such that for ) and for , hence, is strictly increasing in and strictly decreasing in .

From (2.16) and (2.17) together with the monotonicity of we clearly see that there exists such that is strictly increasing in and strictly decreasing in . It follows from (2.13) and (2.14) together with the monotonicity of that there exists such that is strictly increasing in and strictly decreasing in . Then (2.7), (2.10) and (2.11) imply that there exists such that is strictly increasing in and strictly decreasing in .

for .

for .

Therefore, inequality (2.2) follows from (2.3) and (2.4) together with (2.27).

Secondly, we prove that is the best possible upper convex combination bound of arithmetic and harmonic means for the Seiffert's mean .

for .

Therefore, for follows from (2.28) and (2.34).

Finally, we prove that is the best possible lower convex combination bound of arithmetic and harmonic means for the Seiffert's mean .

Inequality (2.35) implies that for any there exists such that for .

## Declarations

### Acknowledgments

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

## Authors’ Affiliations

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