# On Logarithmic Convexity for Ky-Fan Inequality

- Matloob Anwar
^{1}Email author and - J. Pečarić
^{1, 2}

**2008**:870950

**DOI: **10.1155/2008/870950

© M. Anwar and J. Pečarić. 2008

**Received: **19 November 2007

**Accepted: **14 February 2008

**Published: **5 March 2008

## Abstract

We give an improvement and a reversion of the well-known Ky-Fan inequality as well as some related results.

## 1. Introduction and Preliminaries

It is well known that Ky-Fan inequality can be obtained from the Levinson inequality [1], see also [2, page 71].

Theorem 1.1.

In [3], the second author proved the following result.

Theorem 1.2.

In this paper, we will give an improvement and reversion of Ky-Fan inequality as well as some related results.

## 2. Main Results

Lemma 2.1.

Then that is, is 3-convex for .

Theorem 2.2.

for as in (1.2). Then

that is, is log convex in the Jensen sense;

where , .

- (1)Let us consider the function(26)

- (2)Note that is continuous at all points , and since(29)

which completes the proof.

Corollary 2.3.

Proof.

Remark 2.4.

In Corollary 2.3, putting we get an improvement of Ky-Fan inequality.

Theorem 2.5.

for as for Theorem 1.1. Then

that is, is log convex in the Jensen sense;

where , .

Proof.

The proof is similar to the proof of Theorem 2.2.

Remark 2.6.

Let us note that similar results for difference of power means were recently obtained by Simic in [4].

## Authors’ Affiliations

## References

- Levinson N:
**Generalization of an inequality of Ky-Fan.***Journal of Mathematical Analysis and Applications*1964,**8**(1):133–134. 10.1016/0022-247X(64)90089-7MATHMathSciNetView ArticleGoogle Scholar - Pečarić J, Proschan F, Tong YL:
*Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering*.*Volume 187*. Academic Press, Boston, Mass, USA; 1992:xiv+467.MATHGoogle Scholar - Pečarić J:
**An inequality for 3-convex functions.***Journal of Mathematical Analysis and Applications*1982,**90**(1):213–218. 10.1016/0022-247X(82)90055-5MATHMathSciNetView ArticleGoogle Scholar - Simic S:
**On logarithmic convexity for differences of power means.***Journal of Inequalities and Applications*2007,**2007:**-8.Google Scholar

## 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.