Estimations of f- and Rényi Divergences by Using a Cyclic Refinement of the Jensen’s Inequality

Abstract

The Jensen’s inequality plays a crucial role to obtain inequalities for divergences between probability distributions. In this paper, we introduce a new functional, based on the f-divergence functional, and then, we obtain some estimates for the new functional, the f-divergence and the Renyi divergence by applying a cyclic refinement of the Jensen’s inequality. Some inequalities for Renyi and Shannon entropies are obtained too. Zipf–Mandelbrot law is used to illustrate the results.