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

2026-05-18

AI Summary

This study investigates the structural properties of Sharma–Mittal entropy on the lattice of n-dimensional probability distributions defined by the vector majorization order, with a focus on its subadditivity and supermodularity. By integrating majorization theory, lattice theory, and information-theoretic entropy analysis—augmented with convexity arguments and inequality techniques—the work establishes, for the first time, a rigorous proof that this generalized entropy simultaneously satisfies both subadditivity and supermodularity on the said lattice. These findings unify and extend known structural results for Shannon, Tsallis, and Rényi entropies, offering a novel theoretical perspective on entropy functionals in information theory and statistical physics.

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 Rényi entropy.

Problem

• Sharma-Mittal entropy
• subadditivity
• supermodularity
• majorization lattice
• probability distributions

Innovation

• Sharma-Mittal entropy
• subadditivity
• supermodularity
• majorization lattice
• generalized entropy