n-convexity and weighted majorization with applications to f-divergences and Zipf–Mandelbrot law

Abstract

In this paper we obtain refinement of Sherman’s generalization of classical majorization inequality for convex functions (2-convex functions). Using some nice properties of Green’s functions we introduce new identities that include Sherman’s difference, deduced from Sherman’s inequality, which enable us to extend Sherman’s results to the class of convex functions of higher order, i.e. to n-convex functions (\(n\ge 3\)). We connect this approach with Csiszár f-divergence and specified divergences as the Kullback–Leibler divergence, Hellinger divergence, Harmonic divergence, Bhattacharya distance, Triangular discrimination, Rényi divergence and derive new estimates for them. We also observe results in the context of the Zipf–Mandelbrot law and its special form Zipf’s law and give one linguistic example using experimentally obtained values of coefficients from Zipf’s law assigned to different languages.