Estimation of Jensen's gap through an integral identity with applications to divergence

Abstract

Jensen's inequality and results related to its gap play a crucial role in the literature of applied mathematics. In this paper, we introduce a new, simple and sharp bound for Jensen's gap in discrete form by using an integral identity in terms of a certain function. We demonstrate this bound in integral form as well. Also, we illustrate some numerical examples which exhibit the tightness of the bound. The examples show that the bound is better than the existing bound given in [8]. Furthermore, we derive a new bound for the gap of Hermite-Hadamard inequality and derive two new variants of the Holder inequality using the main results. At the end, we obtain some new inequalities for various divergences by using the main result.