부산대학교 도서관

In this paper, the author obtains generalizations and improved resultsrelated to Jensen's inequality, Jessen's inequality, their conversesand Jensen-type interpolating inequalities.\par First, the author provides a generalization of an interpolatinginequality involving positive linear functionals for strongly convexfunctions. From that result, an improvement of Jessen's inequality forstrongly convex functions is derived. Additionally, if the consideredlinear class of functions, denoted as $L$, is equipped with the latticeproperty, an improvement of the converse Jessen inequality is obtained.As a consequence, improvements to the Jensen inequality and itsconverse for strongly convex functions are also derived. Thesegeneralizations are presented in both the continuous and discretecases.\par The main results obtained are applied to the so-called strongly$f$-divergences, a recently introduced concept of $f$-divergences forstrongly convex functions $f$. As an outcome, stronger estimates forsome well-known divergences such as the Kullback@-Leibler divergence,the $\chi^2$-divergence, the Hellinger divergence, the Bhattacharyadistance and the Jeffreys distance are derived.