- Research Article
- Open Access

# Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means

- Bo-Yong Long
^{1, 2}and - Yu-Ming Chu
^{3}Email author

**2010**:905679

https://doi.org/10.1155/2010/905679

© B.-Y. Long and Y.-M. Chu. 2010

**Received:**13 November 2009**Accepted:**25 February 2010**Published:**8 March 2010

## Abstract

For , the power mean of order of two positive numbers and is defined by , for , and , for . In this paper, we answer the question: what are the greatest value and the least value such that the double inequality holds for all and with ? Here , , and denote the classical arithmetic, geometric, and harmonic means, respectively.

## Keywords

- Simple Computation
- Sharp Bound
- Double Inequality

## 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 literatures [1–12]. It is well known that is continuous and increasing with respect to for fixed and . Let , and be the classical arithmetic, geometric, and harmonic means of two positive numbers and , respectively. Then

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

In [15], Mao proved

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

The following sharp bounds for and in terms of power mean are proved in [16]:

The purpose of this paper is to answer the question: what are the greatest value and the least value such that the double inequality

## 2. Main Result

In order to establish our main results we need the following lemma.

Lemma 2.1.

Proof.

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

Therefore, Lemma 2.1(2) follows from (2.1)–(2.3) and (2.5) together with (2.8).

Theorem 2.2.

(2) for , and for , each equality occurs if and only if , and and are the best possible power mean bounds for the product .

( ) If and , then we clearly see that

If and , without loss of generality, we assume that . Let and , then , and simple computations lead to

Therefore, for follows from (2.11) and Lemma 2.1(1) together with (2.12), and for follows from (2.11) and Lemma 2.1(2) together with (2.12).

Next, we prove that and are the best possible power mean bounds for the product .

Firstly, we prove that is the best possible upper power mean bound for the product if .

Let , then the Taylor expansion leads to

Equations (2.13) and (2.14) imply that if , then for any there exists , such that for .

Secondly, we prove that is the best possible lower power mean bound for the product if .

From (2.15) and , we clearly see that

Equation (2.16) implies that if , then for any there exists , such that for .

Thirdly, we prove that is the best possible lower power mean bound for the product if .

Let , then the Taylor expansion leads to

Equations (2.17) and (2.18) imply that if , then for any there exists , such that for .

Finally, we prove that is the best possible upper power mean bound for the product if .

From (2.19) and we clearly see that

Equation (2.20) implies that if , then for any there exists , such that for .

## Declarations

### Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (Grant no. 60850005) and the Natural Science Foundation of Zhejiang Province (Grant no. D7080080, Y607128).

## Authors’ Affiliations

## References

- Wu S: 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.2005.03.050MathSciNetView ArticleMATHGoogle Scholar - Richards 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.2005.01.018MathSciNetView ArticleMATHGoogle Scholar - Wang WL, Wen JJ, Shi HN: Optimal inequalities involving power means.
*Acta Mathematica Sinica*2004, 47(6):1053–1062.MathSciNetMATHGoogle Scholar - Hästö PA: Optimal inequalities between Seiffert's mean and power means.
*Mathematical Inequalities & Applications*2004, 7(1):47–53.MathSciNetView ArticleMATHGoogle Scholar - Alzer H: A power mean inequality for the gamma function.
*Monatshefte für Mathematik*2000, 131(3):179–188. 10.1007/s006050070007MathSciNetView ArticleMATHGoogle Scholar - Alzer H, Qiu S-L: Inequalities for means in two variables.
*Archiv der Mathematik*2003, 80(2):201–215. 10.1007/s00013-003-0456-2MathSciNetView ArticleMATHGoogle Scholar - Tarnavas CD, Tarnavas DD: An inequality for mixed power means.
*Mathematical Inequalities & Applications*1999, 2(2):175–181.MathSciNetView ArticleMATHGoogle Scholar - Bukor J, Tóth J, Zsilinszky L: The logarithmic mean and the power mean of positive numbers.
*Octogon Mathematical Magazine*1994, 2(1):19–24.MathSciNetGoogle Scholar - 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)90339-2MathSciNetView ArticleMATHGoogle Scholar - Chen J, Hu B: The identric mean and the power mean inequalities of Ky Fan type.
*Facta Universitatis. Series: Mathematics and Informatics*1989, (4):15–18.MATHGoogle Scholar - Imoru CO: The power mean and the logarithmic mean.
*International Journal of Mathematics and Mathematical Sciences*1982, 5(2):337–343. 10.1155/S0161171282000313MathSciNetView ArticleMATHGoogle Scholar - Lin TP: The power mean and the logarithmic mean.
*The American Mathematical Monthly*1974, 81: 879–883. 10.2307/2319447MathSciNetView ArticleMATHGoogle Scholar - Alzer H, Janous W: Solution of problem .
*Crux Mathematicorum*1987, 13: 173–178.Google Scholar - Bullen PS, Mitrinović DS, Vasić PM:
*Means and Their Inequalities, Mathematics and Its Applications*.*Volume 31*. D. Reidel, Dordrecht, The Netherlands; 1988:xx+459.View ArticleGoogle Scholar - Mao Q-J: Power mean, logarithmic mean and Heronian dual mean of two positive number.
*Journal of Suzhou College of Education*1999, 16(1–2):82–85.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

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