On further refinements of the Jensen inequality and applications

Abstract

We aim to extend some recent refinements of the Jensen inequality. First, we establish even more precise estimates for the Jensen inequality. Then, we derive an improvement of the Jensen inequality with regard to ‐convexity. As an application, our main results are applied in deriving refinements of power mean inequalities and the Hölder inequality. Finally, we obtain more accurate estimates for ‐divergences, important in the information theory.