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On Logarithmic Convexity for Ky-Fan Inequality
Journal of Inequalities and Applications volume 2008, Article number: 870950 (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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ4_HTML.gif)
Then that is,
is 3-convex for
.
Theorem 2.2.
Define the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ5_HTML.gif)
for as in (1.2). Then
(1)for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ6_HTML.gif)
that is, is log convex in the Jensen sense;
(2) is continuous on
, it is also log convex, that is, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ7_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ8_HTML.gif)
where ,
.
Proof.
-
(1)
Let us consider the function
(26)
where ,
are reals.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ10_HTML.gif)
for . This implies that
is 3-convex. Therefore, by (1.2), we have
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ11_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ13_HTML.gif)
which completes the proof.
Corollary 2.3.
For ,
as in (1.2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ14_HTML.gif)
Proof.
Setting , and
in Theorem 1.2, we get
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ15_HTML.gif)
Again setting , and
in Theorem 1.2, we get
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ16_HTML.gif)
Combining both inequalities (2.12), (2.13), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ17_HTML.gif)
Also we have positive for
; therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ18_HTML.gif)
Applying exponentional function, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ19_HTML.gif)
Remark 2.4.
In Corollary 2.3, putting we get an improvement of Ky-Fan inequality.
Theorem 2.5.
Define the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ20_HTML.gif)
for as for Theorem 1.1. Then
(1)for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ21_HTML.gif)
that is, is log convex in the Jensen sense;
(2) is continuous on
, it is also log convex. That is for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ22_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F870950/MediaObjects/13660_2007_Article_1872_Equ23_HTML.gif)
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 Ky-Fan. Journal of Mathematical Analysis and Applications 1964,8(1):133–134. 10.1016/0022-247X(64)90089-7
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 3-convex functions. Journal of Mathematical Analysis and Applications 1982,90(1):213–218. 10.1016/0022-247X(82)90055-5
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 Ky-Fan 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