부산대학교 도서관

In this paper, the author obtains generalizations and improved results related to Jensen's inequality, Jessen's inequality, their converses and Jensen-type interpolating inequalities. \par First, the author provides a generalization of an interpolating inequality involving positive linear functionals for strongly convex functions. From that result, an improvement of Jessen's inequality for strongly convex functions is derived. Additionally, if the considered linear class of functions, denoted as $L$, is equipped with the lattice property, an improvement of the converse Jessen inequality is obtained. As a consequence, improvements to the Jensen inequality and its converse for strongly convex functions are also derived. These generalizations are presented in both the continuous and discrete cases. \par The main results obtained are applied to the so-called strongly $f$-divergences, a recently introduced concept of $f$-divergences for strongly convex functions $f$. As an outcome, stronger estimates for some well-known divergences such as the Kullback@-Leibler divergence, the $\chi^2$-divergence, the Hellinger divergence, the Bhattacharya distance and the Jeffreys distance are derived.