Refinement of the Jensen integral inequality

Abstract In this paper we give a refinement of Jensen’s integral inequality and its generalization for linear functionals. We also present some applications in Information Theory.


Introduction
Let C be a convex subset of the linear space X and f be a convex function on C. If p D .p 1 ; :::; p n / is probability sequence and x D .x 1 ; :::; x n / 2 C n , then is well known in the literature as Jensen's inequality. The Lebesgue integral version of the Jensen inequality is given below: In case when is strictly convex on I one has equality in .2/ if and only if f is constant almost everywhere on .
The Jensen inequality for convex functions plays a crucial role in the Theory of Inequalities due to the fact that other inequalities such as the arithmetic mean-geometric mean inequality, the Hölder and Minkowski inequalities, the Ky Fan inequality etc. can be obtained as particular cases of it. There is an extensive literature devoted to Jensen's inequality concerning different generalizations, refinements, counterparts and converse results, see, for example [1][2][3][4][5][6][7][8][9].
In this paper we give a refinement of Jensen's integral inequality and its generalization for linear functionals. We also present some applications in Information Theory for example for Kullback-Leibler, total variation and Karl Pearson 2 -divergences etc. Let . ; ƒ; / be a measure space with 0 < . / < 1 and L. ; ƒ; / D ff W ! R W f is measurable and R f .t /d .t / < 1g be a Lebesgue space. Consider the set S D f! 2 ƒ W .!/ ¤ 0 and .!/ D . n !/ ¤ 0g and W .a; b/ ! R be a convex function defined on an open interval .a; b/. If f W ! .a; b/ is such that f; ı f 2 L. ; ƒ; /, then for any set ! 2 S, define the functional as We give the following refinement of Jensen's inequality.
Proof. As for any ! 2 S we have Therefore by the convexity of the function we get Also for any ! 2 S and by the Jensen inequality we have From (5) and (6) As a simple consequence of Theorem 2.1 we can obtain refinement of Hermite-Hadamard inequality:

Further generalization
Let E be a nonempty set, A be an algebra of subsets of E, and L be a linear class of real-valued functions f W E ! R having the properties: L1 : f; g 2 L ) .˛f Cˇg/ 2 L for all˛;ˇ2 R; L2 : where E 1 is the indicator function of E 1 . It follows from L 2 ; L 3 that E 1 2 L for every E 1 2 A. A positive isotonic linear functional A W L ! R is a functional satisfying the following properties: Jessen (see [10, p-47]) gave the following generalization of Jensen's inequality for convex functions.
The following refinement of (8) holds.
Using the inequality (8) we obtain This proves the second inequality in (9). The first inequality follows by using definition of convex function and identity (7).

Applications for Csiszár divergence measures
Let . ; ƒ; / be a probability measure space. Consider the set of all density functions on to be S WD fpjp W ! R; p.s/ > 0; R p.s/d .s/ D 1g. Csiszár introduced the concept of f divergence for a convex function f W .0; 1/ ! . 1; 1/ (cf. [11] , see also [12]) by  One parametric generalization of the Kullback-Leibler [13] relative information was studied in a different way by Cressie and Read [14].
This class gives, for particular values of˛; some important divergences. For instance, for˛D 1 2 it provides a distance, namely, the Hellinger distance.
There are various other divergences in Information Theory and Statistics such as Arimoto-type divergences, Matushita's divergence, Puri-Vincze divergences etc. ( cf. [15], [16]) used in various problems in Information Theory and statistics. An application of Theorem 1.1 is the following result given by Csiszár and Korner (cf. [17]).
Proof. By putting f .x/ D jx 1j for all x 0 in Theorem 4.2 we get (14). . 0/: Proof. By putting f .t / D t ln.t / in Theorem 4.2 one can get first inequality in (16).
To prove the second inequality, we utilize the inequality between the geometric mean and harmonic mean, x; y;˛2 OE0; 1; we have for Proof. By putting f .x/ D x˛for˛> 1; x > 0; in Theorem 4.2 we get the required inequalities.