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

• DOI: 10.1007/s40840-017-0526-4
• Publication Date: 2017-07-28T08:49:02Z
• Authors: László Horváth; Đilda Pečarić; Josip Pečarić
• Keywords: Divergence (linguistics); Jensen's inequality; Zipf's law; Mandelbrot set

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 Rényi divergence by applying a cyclic refinement of the Jensen’s inequality. Some inequalities for Rényi and Shannon entropies are obtained too. Zipf–Mandelbrot law is used to illustrate the results.