Bounds for some entropies and special functions

TL;DR

This paper derives bounds for certain entropies related to a family of probability distributions, including binomial, Poisson, and negative binomial, by analyzing the index of coincidence and its differential equation.

Contribution

It introduces new bounds for the index of coincidence and associated entropies using properties of the Heun differential equation and convexity analysis.

Findings

Bounds for the index of coincidence are established.

Bounds for Rényi and Tsallis entropies of order 2 are derived.

The index of coincidence is shown to be a logarithmically convex function.

Abstract

We consider a family of probability distributions depending on a real parameter and including the binomial, Poisson and negative binomial distributions. The corresponding index of coincidence satisfies a Heun differential equation and is a logarithmically convex function. Combining these facts we get bounds for the index of coincidence, and consequently for R\'{e}nyi and Tsallis entropies of order $2$.