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

- Yu-Ming Chu
^{1}Email author and - Wei-Feng Xia
^{2}

**2009**:741923

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

© Y.-M. Chu and W.-F. Xia. 2009

**Received: **23 July 2009

**Accepted: **30 October 2009

**Published: **22 November 2009

## Abstract

## 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]):

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.

Proof.

If , then we clearly see that .

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

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

Remark 2.2.

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

Theorem 2.3.

Proof.

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

Remark 2.4.

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

## Declarations

### 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.

## Authors’ Affiliations

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