Improvements of the inequalities for the f-divergence functional with applications to the Zipf-Mandelbrot law

Butt, Saad Ishan and Horváth, László and Pečarić, Gilda and Pečarić, Josip (2019) Improvements of the inequalities for the f-divergence functional with applications to the Zipf-Mandelbrot law. In: Inequalities and Zipf-Mandelbrot Law. Element, pp. 167-203. ISBN 978-953-197-670-1

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Abstract

The Jensen’s inequality plays a crucial role to obtain inequalities for divergences between probability distributions. In this chapter, 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.

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