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

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.