Jensen--Mercer-type Inequalities and Strongly $f$-Divergences

Abstract

In this paper, we develop refined Jensen--Mercer-type inequalities for strongly convex functions, including reverse inequalities in the differentiable case. We identify the relationships between these inequalities and known bounds. Our method exploits a structural decomposition of strongly convex functions together with classical convexity arguments. As applications, we obtain explicit and computable bounds for strongly $f$-divergences, leading to new quantitative estimates for major divergence measures such as the Kullback--Leibler divergence, $χ^{2}$-divergence, Hellinger divergence, Bhattacharyya-type divergence, and Jeffreys divergence.