- Research Article
- Open access
- Published:

# Two Sharp Inequalities for Power Mean, Geometric Mean, and Harmonic Mean

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 741923 (2009)

## Abstract

For , the power mean of order of two positive numbers and is defined by . In this paper, we establish two sharp inequalities as follows: and for all . Here and denote the geometric mean and harmonic mean of and respectively.

## 1. Introduction

For , the power mean of order of two positive numbers and is defined by

Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for can be found in literature [1–12]. It is well known that is continuous and increasing with respect to for fixed and . If we denote by and the arithmetic mean, geometric mean and harmonic mean of and , respectively, then

In [13], Alzer and Janous established the following sharp double-inequality (see also [14,page 350]):

for all

In [15], Mao proved

for all , and is the best possible lower power mean bound for the sum .

The purpose of this paper is to answer the questions: what are the greatest values and , and the least values and , such that and for all ?

## 2. Main Results

Theorem 2.1.

for all , equality holds if and only if , and is the best possible lower power mean bound for the sum .

Proof.

If , then we clearly see that .

If and , then simple computation leads to

Next, we prove that is the best possible lower power mean bound for the sum .

For any and , one has

where .

Let , then the Taylor expansion leads to

Equations (2.2) and (2.3) imply that for any there exists , such that for .

Remark 2.2.

For any , one has

Therefore, is the best possible upper power mean bound for the sum .

Theorem 2.3.

for all , equality holds if and only if , and is the best possible lower power mean bound for the sum .

Proof.

If , then we clearly see that

If and , then elementary calculation yields

Next, we prove that is the best possible lower power mean bound for the sum .

For any and , one has

where .

Let , then the Taylor expansion leads to

Equations (2.6) and (2.7) imply that for any there exists , such that

for .

Remark 2.4.

For any , one has

Therefore, is the best possible upper power mean bound for the sum .

## References

Wu SH:

**Generalization and sharpness of the power means inequality and their applications.***Journal of Mathematical Analysis and Applications*2005,**312**(2):637–652. 10.1016/j.jmaa.2005.03.050Richards KC:

**Sharp power mean bounds for the Gaussian hypergeometric function.***Journal of Mathematical Analysis and Applications*2005,**308**(1):303–313. 10.1016/j.jmaa.2005.01.018Wang WL, Wen JJ, Shi HN:

**Optimal inequalities involving power means.***Acta Mathematica Sinica*2004,**47**(6):1053–1062.Hästö PA:

**Optimal inequalities between Seiffert's mean and power means.***Mathematical Inequalities & Applications*2004,**7**(1):47–53.Alzer H, Qiu S-L:

**Inequalities for means in two variables.***Archiv der Mathematik*2003,**80**(2):201–215. 10.1007/s00013-003-0456-2Alzer H:

**A power mean inequality for the gamma function.***Monatshefte für Mathematik*2000,**131**(3):179–188. 10.1007/s006050070007Tarnavas CD, Tarnavas DD:

**An inequality for mixed power means.***Mathematical Inequalities & Applications*1999,**2**(2):175–181.Bukor J, Tóth J, Zsilinszky L:

**The logarithmic mean and the power mean of positive numbers.***Octogon Mathematical Magazine*1994,**2**(1):19–24.Pečarić JE:

**Generalization of the power means and their inequalities.***Journal of Mathematical Analysis and Applications*1991,**161**(2):395–404. 10.1016/0022-247X(91)90339-2Chen J, Hu B:

**The identric mean and the power mean inequalities of Ky Fan type.***Facta Universitatis*1989, (4):15–18.Imoru CO:

**The power mean and the logarithmic mean.***International Journal of Mathematics and Mathematical Sciences*1982,**5**(2):337–343. 10.1155/S0161171282000313Lin TP:

**The power mean and the logarithmic mean.***The American Mathematical Monthly*1974,**81:**879–883. 10.2307/2319447Alzer H, Janous W:

**Solution of problem 8*.***Crux Mathematicorum*1987,**13:**173–178.Bullen PS, Mitrinović DS, Vasić PM:

*Means and Their Inequalities, Mathematics and Its Applications (East European Series)*.*Volume 31*. D. Reidel, Dordrecht, The Netherlands; 1988:xx+459.Mao QJ:

**Power mean, logarithmic mean and Heronian dual mean of two positive number.***Journal of Suzhou College of Education*1999,**16**(1–2):82–85.

## Acknowledgments

This research is partly supported by N S Foundation of China under Grant 60850005 and the N S Foundation of Zhejiang Province under Grants Y7080185 and Y607128.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Chu, YM., Xia, WF. Two Sharp Inequalities for Power Mean, Geometric Mean, and Harmonic Mean.
*J Inequal Appl* **2009**, 741923 (2009). https://doi.org/10.1155/2009/741923

Received:

Accepted:

Published:

DOI: https://doi.org/10.1155/2009/741923