Convexity Properties of Some Entropies (II)

This is a continuation of the author’s paper “Convexity properties of some entropies”, published in Raşa (Results Math 73:105, 2018). We consider the sum Fn(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_n(x)$$\end{document} of the squared fundamental Bernstein polynomials of degree n, in relation with Rényi entropy and Tsallis entropy for the binomial distribution with parameters n and x. Several functional equations and inequalities for these functions are presented. In particular, we give a new and simpler proof of a conjecture asserting that Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_n$$\end{document} is logarithmically convex. New combinatorial identities are obtained as a byproduct. Rényi entropies and Tsallis entropies for more general families of probability distributions are considered. The paper ends with three new conjectures.