# Inequalities for Generalized Logarithmic Means

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

**2009**:763252

https://doi.org/10.1155/2009/763252

© Y.-M. Chu and W.-F. Xia. 2009

**Received: **2 June 2009

**Accepted: **10 December 2009

**Published: **24 January 2010

## Abstract

For , the generalized logarithmic mean of two positive numbers and is defined as , for , , for , , , , for , , and , for , . In this paper, we prove that , and for all , and the constants , and cannot be improved for the corresponding inequalities. Here , and denote the arithmetic, geometric, and harmonic means of and , respectively.

## Keywords

## 1. Introduction

In [1], the following results are established: (1) implies that ; (2) implies that ; (3) implies that there exist such that ; (4) implies that there exist such that . Hence the question was answered: what are the least value and the greatest value such that the inequality holds for all ?

Stolarsky [2] proved that , with equality if and only if .

Here , are sharp and equality holds only if or or . The case reduces to Lin's results [1]. Other generalizations of Lin's results were given by Imoru [4].

for all , and . The upper bound in (1.6) is the best possible.

for all , where the function is the logarithmic derivative of the gamma function.

Recently, some monotonicity results of the ratio between generalized logarithmic means were established in [7–9].

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

## 2. Main Results

Theorem 2.1.

Proof.

Equations (2.3) and (2.4) imply that for any there exists such that for .

Theorem 2.2.

Proof.

Next we prove that is the optimal value for which the inequality holds.

Equations (2.7) and (2.8) imply that for any there exists such that for .

Theorem 2.3.

Proof.

Equations (2.10) and (2.11) imply that for any there exists such that for .

Remark 2.4.

Therefore, we cannot get inequality for any and all

Remark 2.5.

## Declarations

### Acknowledgments

This research is partly supported by N S Foundation of China under Grant 60850005 and N S Foundation of Zhejiang Province under Grants D7080080 and Y7080185.

## Authors’ Affiliations

## References

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## 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.