A refinement of the discrete Jensen's inequality

Abstract

Abstract.We give a refinement of the discrete Jensen's inequality in the convex and mid-convex cases. For mid-convex functions our result is a common generalization of known inequalities. We illustrate the scope of the results by applying them to some special situations.Mathematics subject classification (2010): 26D07 (26A51).

A parameter-dependent refinement of the discrete Jensen's inequality for convex and mid-convex functions

“…In a recent work, [4] Horváth and Pečarić define a lot of new sequences, they generalize and give a uniform treatment a number of wellknown results from this area, especially (5) and (6) are extended. Horváth develops a method in [5] to construct decreasing real sequences satisfying (3).…”

“…Horváth develops a method in [5] to construct decreasing real sequences satisfying (3). His paper contains some improvements of the results in [4] and gives a new approach of the topic. The description of the sequences in [4,5] requires some work, so we do not go into the details.…”

“…His paper contains some improvements of the results in [4] and gives a new approach of the topic. The description of the sequences in [4,5] requires some work, so we do not go into the details. The problem (B) has been considered for the classical Jensen's inequality by Horváth [6].…”

Estimation of different entropies via Lidstone polynomial using Jensen-type functionals

“…For many years, Jensen's inequality has been of great interest. It was refined by defining some new functions (see [14,15]). Horváth and Pečarić ([12,15], see also [13, p. 26]) gave a refinement of Jensen's inequality for convex function.…”

“…It was refined by defining some new functions (see [14,15]). Horváth and Pečarić ([12,15], see also [13, p. 26]) gave a refinement of Jensen's inequality for convex function. They defined some essential notions to prove the refinement given as follows: Let X be a set, and: P(X ) := Power set of X , |X |:= Number of elements of X , N:= Set of natural numbers with 0.…”