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On Logarithmic Convexity for KyFan Inequality
Journal of Inequalities and Applications volume 2008, Article number: 870950 (2008)
Abstract
We give an improvement and a reversion of the wellknown KyFan 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 abovementioned means , and . Then the wellknown KyFan inequality is
It is well known that KyFan inequality can be obtained from the Levinson inequality [1], see also [2, page 71].
Theorem 1.1.
Let be a realvalued 3convex function on , then for
In [3], the second author proved the following result.
Theorem 1.2.
Let be a realvalued 3convex function on and points on , then
In this paper, we will give an improvement and reversion of KyFan inequality as well as some related results.
2. Main Results
Lemma 2.1.
Define the function
Then that is, is 3convex for .
Theorem 2.2.
Define the function
for as in (1.2). Then
(1)for all ,
that is, is log convex in the Jensen sense;
(2) is continuous on , it is also log convex, that is, for ,
with
where , .
Proof.

(1)
Let us consider the function
(26)
where , are reals.
for . This implies that is 3convex. Therefore, by (1.2), we have , that is,
This follows that is log convex in the Jensen sense.

(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,
which completes the proof.
Corollary 2.3.
For , as in (1.2),
Proof.
Setting , and in Theorem 1.2, we get or
Again setting , and in Theorem 1.2, we get or
Combining both inequalities (2.12), (2.13), we get
Also we have positive for ; therefore, we have
Applying exponentional function, we get
Remark 2.4.
In Corollary 2.3, putting we get an improvement of KyFan inequality.
Theorem 2.5.
Define the function
for as for Theorem 1.1. Then
(1)for all ,
that is, is log convex in the Jensen sense;
(2) is continuous on , it is also log convex. That is for ,
with
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
Levinson N: Generalization of an inequality of KyFan. Journal of Mathematical Analysis and Applications 1964,8(1):133–134. 10.1016/0022247X(64)900897
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.
Pečarić J: An inequality for 3convex functions. Journal of Mathematical Analysis and Applications 1982,90(1):213–218. 10.1016/0022247X(82)900555
Simic S: On logarithmic convexity for differences of power means. Journal of Inequalities and Applications 2007, 2007:8.
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Anwar, M., Pečarić, J. On Logarithmic Convexity for KyFan Inequality. J Inequal Appl 2008, 870950 (2008). https://doi.org/10.1155/2008/870950
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DOI: https://doi.org/10.1155/2008/870950