Open Access

On Logarithmic Convexity for Ky-Fan Inequality

Journal of Inequalities and Applications20082008:870950

DOI: 10.1155/2008/870950

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

Let and be real numbers such that with . Let and be the weighted geometric mean and arithmetic mean, respectively, defined by , and . In particular, consider the above-mentioned means , and . Then the well-known Ky-Fan inequality is
(11)

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

Theorem 1.1.

Let be a real-valued 3-convex function on , then for
(12)

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

Theorem 1.2.

Let be a real-valued 3-convex function on and points on , then
(13)

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.

Define the function
(21)

Then that is, is 3-convex for .

Theorem 2.2.

Define the function
(22)

for as in (1.2). Then

(1)for all ,
(23)

that is, is log convex in the Jensen sense;

(2) is continuous on , it is also log convex, that is, for ,
(24)
with
(25)

where , .

Proof.
  1. (1)
    Let us consider the function
    (26)
     
where , are reals.
(27)
for . This implies that is 3-convex. Therefore, by (1.2), we have , that is,
(28)
This follows that is log convex in the Jensen sense.
  1. (2)
    Note that is continuous at all points , and since
    (29)
     
Since is a continuous and convex in Jensen sense, it is log convex. That is,
(210)

which completes the proof.

Corollary 2.3.

For , as in (1.2),
(211)

Proof.

Setting , and in Theorem 1.2, we get or
(212)
Again setting , and in Theorem 1.2, we get or
(213)
Combining both inequalities (2.12), (2.13), we get
(214)
Also we have positive for ; therefore, we have
(215)
Applying exponentional function, we get
(216)

Remark 2.4.

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

Theorem 2.5.

Define the function
(217)

for as for Theorem 1.1. Then

(1)for all ,
(218)

that is, is log convex in the Jensen sense;

(2) is continuous on , it is also log convex. That is for ,
(219)
with
(220)

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

(1)
Abdus Salam School of Mathematical Sciences, GC University
(2)
Faculty of Textile Technology, University of Zagreb

References

  1. 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
  2. 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
  3. 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
  4. Simic S: On logarithmic convexity for differences of power means. Journal of Inequalities and Applications 2007, 2007:-8.Google Scholar

Copyright

© M. Anwar and J. Pečarić. 2008

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.