Bounds for some entropies and special functions

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 Rényi and Tsallis entropies of order $2$.

Accepted to be published in Carpathian Journal of Mathematics, 1/2018