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# On Logarithmic Convexity for Ky-Fan Inequality

## 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].

## 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)90089-7

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.

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)90055-5

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

## Author information

Authors

### Corresponding author

Correspondence to Matloob Anwar.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Anwar, M., Pečarić, J. On Logarithmic Convexity for Ky-Fan Inequality. J Inequal Appl 2008, 870950 (2008). https://doi.org/10.1155/2008/870950