- Research Article
- Open Access
On Some Improvements of the Jensen Inequality with Some Applications
© M. Adil Khan et al. 2009
Received: 23 April 2009
Accepted: 10 August 2009
Published: 27 August 2009
An improvement of the Jensen inequality for convex and monotone function is given as well as various applications for mean. Similar results for related inequalities of the Jensen type are also obtained. Also some applications of the Cauchy mean and the Jensen inequality are discussed.
The well-known Jensen's inequality for convex function is given as follows.
Here and in the whole paper we suppose that all integrals exist. By considering the difference of (1.1) for functional in  Anwar and Pečarić proved an interesting result of log-convexity. We can define this result for integrals as follows.
The following improvement of (1.1) was obtained in .
In this paper, we give another proof and extension of Theorem 1.2 as well as improvements of Theorem 1.3 for monotone convex function with some applications. Also we give applications of the Jensen inequality for divergence measures in information theory and related Cauchy means.
2. Another Proof and Extension of Theorem 1.2
In fact, Theorem 1.2 for and was first of all initiated by Simić in .
In  we have given correct proof by using extension of (2.1), so that it is defined on .
Moreover, we can give another proof so that we use only (2.1) but without using convexity as in .
Proof of Theorem 1.2.
Consider the function defined, as in , by (2.1).
Let us note that it was used in  to get corresponding Cauchy's means. Moreover, we can extend the above result.
3. Improvements of the Jensen Inequality for Monotone Convex Function
Now from (1.4), (3.3), and (3.4) we get (3.1).
Of course a discrete inequality is a simple consequence of Theorem 3.1.
The following improvement of the Hermite-Hadamard inequality is valid .
4. Improvements of the Levinson Inequality
As for -convex function the function is convex on , so by setting in the discrete case of [2, Theorem 2], we get (4.1).
Ky Fan Inequality
Inequality (4.5) has evoked the interest of several mathematicians and in numerous articles new proofs, extensions, refinements and various related results have been published .
The following improvement of Ky Fan inequality is valid .
From (4.9) we get (4.7).
5. On Some Inequalities for Csiszár Divergence Measures
Let be a measure space satisfying and a -finite measure on with values in . Let be the set of all probability measures on the measurable space which are absolutely continuous with respect to . For , let and denote the Radon-Nikodym derivatives of and with respect to respectively.
There are various other divergences in Information Theory and statistics such as Arimoto-type divergences, Matushita's divergence, Puri-Vincze divergences (cf. [12–14]) used in various problems in Information Theory and statistics. An application of Theorem 1.1 is the following result given by Csiszár and Körner (cf. ).
Similar consequence of Theorems 1.2 and 2.1 in information theory for divergence measures discussed above is the following result.
As we said in  we define new means of the Cauchy type, here we define an application of these means for divergence measures in the following definition.
An application of Theorem 1.3 in divergence measure is the following result given in .
This research work is funded by the Higher Education Commission Pakistan. The research of the fourth author is supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888.
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