- 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

## Keywords

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

## 1. Introduction

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.

We divide the proof into two cases.

Case 1.

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 .

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 .

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

## References

- Seiffert H-J: Problem 887.
*Nieuw Archief voor Wiskunde*1993, 11(2):176–176.MathSciNetGoogle Scholar - Wang M-K, Chu Y-M, Qiu Y-F: Some comparison inequalities for generalized Muirhead and identric means.
*Journal of Inequalities and Applications*2010, 2010:-10.Google Scholar - Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means.
*Journal of Inequalities and Applications*2010, 2010:-10.Google Scholar - Long B-Y, Chu Y-M: Optimal power mean bounds for the weighted geometric mean of classical means.
*Journal of Inequalities and Applications*2010, 2010:-6.Google Scholar - Xia W-F, Chu Y-M, Wang G-D: The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means.
*Abstract and Applied Analysis*2010, 2010:-9.Google Scholar - Chu Y-M, Long B-Y: Best possible inequalities between generalized logarithmic mean and classical means.
*Abstract and Applied Analysis*2010, 2010:-13.Google Scholar - Shi M-Y, Chu Y-M, Jiang Y-P: Optimal inequalities among various means of two arguments.
*Abstract and Applied Analysis*2009, 2009:-10.Google Scholar - Chu Y-M, Xia W-F: Two sharp inequalities for power mean, geometric mean, and harmonic mean.
*Journal of Inequalities and Applications*2009, 2009:-6.Google Scholar - Chu Y-M, Xia W-F: Inequalities for generalized logarithmic means.
*Journal of Inequalities and Applications*2009, 2009:-7.Google Scholar - Wen J-J, Wang W-L: The optimization for the inequalities of power means.
*Journal of Inequalities and Applications*2006, 2006:-25.Google Scholar - Hara T, Uchiyama M, Takahasi S-E: A refinement of various mean inequalities.
*Journal of Inequalities and Applications*1998, 2(4):387–395. 10.1155/S1025583498000253MathSciNetMATHGoogle Scholar - Neuman E, Sándor J: On the Schwab-Borchardt mean. II.
*Mathematica Pannonica*2006, 17(1):49–59.MathSciNetMATHGoogle Scholar - Hästö PA: Optimal inequalities between Seiffert's mean and power means.
*Mathematical Inequalities & Applications*2004, 7(1):47–53.MathSciNetView ArticleMATHGoogle Scholar - Neuman E, Sándor J: On the Schwab-Borchardt mean.
*Mathematica Pannonica*2003, 14(2):253–266.MathSciNetMATHGoogle Scholar - Neuman E, Sándor J: On certain means of two arguments and their extensions. International Journal of Mathematics and Mathematical Sciences 2003, (16):981–993.Google Scholar
- Hästö PA: A monotonicity property of ratios of symmetric homogeneous means.
*Journal of Inequalities in Pure and Applied Mathematics*2002, 3(5, article 71):1–23.MathSciNetMATHGoogle Scholar - Jagers AA: Solution of problem 887.
*Nieuw Archief voor Wiskunde*1994, 12: 230–231.Google Scholar - Seiffert H-J: Ungleichungen für einen bestimmten Mittelwert.
*Nieuw Archief voor Wiskunde*1995, 13(2):195–198.MathSciNetGoogle Scholar - Sándor J: On certain inequalities for means. III.
*Archiv der Mathematik*2001, 76(1):34–40. 10.1007/s000130050539MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.