The Sharma-Mittal Entropy is Subadditive and Supermodular on the Majorization Lattice

TL;DR

This paper proves that Sharma-Mittal entropy exhibits subadditivity and supermodularity on the probability distribution lattice, extending known properties of Shannon, Tsallis, and Rényi entropies.

Contribution

It establishes the subadditive and supermodular nature of Sharma-Mittal entropy on the majorization lattice, unifying and extending prior entropy results.

Findings

Sharma-Mittal entropy is subadditive on the majorization lattice.

Sharma-Mittal entropy is supermodular on the majorization lattice.

The results unify properties of several classical entropies.

Abstract

We prove that Sharma-Mittal entropy is a subadditive and supermodular function on the lattice of all $n$-dimensional probability distributions, ordered according to the partial order relation defined by majorization among vectors. Our result unifies and extends analogous results presented in the literature for the Shannon entropy, the Tsallis entropy, and the R\'enyi entropy.