- 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.
- Convex Function
- Divergence Measure
- Hellinger Distance
- Renyi Entropy
- Jensen Inequality
The well-known Jensen's inequality for convex function is given as follows.
is valid for any convex function . In the case when is strictly convex on one has equality in (1.1) if and only if is constant almost everywhere on .
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 .
If is concave, then left-hand side of (1.4) should be .
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.
In fact, Theorem 1.2 for and was first of all initiated by Simić in .
where and are real with .
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).
Since we have and , we also have that (2.4) is valid for . So is log-convex function in the Jensen sense on .
Let us note that it was used in  to get corresponding Cauchy's means. Moreover, we can extend the above result.
where define the determinant of order with elements and .
for and and .
Therefore, ( denote the matrix with elements ) is nonnegative semi definite and (2.6) is valid for . Moreover, since we have continuity of for all , (2.6) is valid for all .
In Theorem 2.1, if we set we get Theorem 1.2.
In this section and in the following section, we denote and .
for monotone convex function . If is monotone concave, then the left-hand side of (3.1) should be .
Now from (1.4), (3.3), and (3.4) we get (3.1).
The case when is nonincreasing can be treated in a similar way.
Of course a discrete inequality is a simple consequence of Theorem 3.1.
The following improvement of the Hermite-Hadamard inequality is valid .
Let be a differentiable convex. Then
holds. If is differentiable concave, then the left-hand side of (3.7) should be
holds. If is differentiable and monotone concave then the left-hand side of (3.8) should be .
Setting in (1.4), we get (3.7).
Setting , and in (3.1), we get (3.8).
As for -convex function the function is convex on , so by setting in the discrete case of [2, Theorem 2], we get (4.1).
As is monotone convex, so by setting in (3.5), we get (5.16).
Ky Fan Inequality
with equality sign if and only if .
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 .
Setting , in (4.1), we get (4.6).
From (4.9) we get (4.7).
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.
is called the -divergence of the probability distributions and .
The Dichotomy class: this class is generated by the family of functions ,
where and are positive integrable functions with
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. ).
By substituting and in Theorem 1.1 we get (5.13).
Similar consequence of Theorems 1.2 and 2.1 in information theory for divergence measures discussed above is the following result.
and let be positive. Then
where define the determinant of order with elements and
(ii) is log-convex.
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.
where and .
Also for we consider limiting case and the result follows from continuity of .
An application of Theorem 1.3 in divergence measure is the following result given in .
By substituting and in Theorem 1.3, we get (5.20).
and as in Theorem 5.7.
By substituting and in Theorem 3.1(ii) we get (5.22).
and as in Theorem 5.7.
The proof follows by setting in Theorem 5.7.
Let be as given in (5.11), then
and as in Theorem 5.7.
The proof follows be setting to be as given in (5.11), in Theorem 3.1.
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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