Strong Convexity-Based Improvements of Jensen–Mercer Inequalities in Functional, Integral and Probabilistic Settings

By using recently established refinements of Jessen and converse Jessen-type inequalities, together with improved characterizations of strongly convex functions, we derive new Jensen–Mercer-type functional inequalities. The obtained results extend and sharpen several known inequalities for convex functions. As applications, we establish new integral inequalities which improve some recently published estimates. In addition, we obtain inequalities involving generalized logarithmic means. Finally, we derive corresponding probabilistic versions of the main results, which lead to refinements of several known results from the literature.