Open Access Research Article

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

Yu-Ming Chu1* and Wei-Feng Xia2

Author Affiliations

1 Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

2 School of Teacher Education, Huzhou Teachers College, Huzhou 313000, China

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Journal of Inequalities and Applications 2009, 2009:741923 doi:10.1155/2009/741923


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2009/1/741923


Received:23 July 2009
Accepted:30 October 2009
Published:22 November 2009

© 2009 The Author(s).

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.

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

(11)

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

(12)

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

(13)

for all

In [15], Mao proved

(14)

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

(21)

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

For any and , one has

(22)

where .

Let , then the Taylor expansion leads to

(23)

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

Remark 2.2.

For any , one has

(24)

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

(25)

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

For any and , one has

(26)

where .

Let , then the Taylor expansion leads to

(27)

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

(28)

for .

Remark 2.4.

For any , one has

(29)

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

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.

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