Majorization precursors to supermodularity and subadditivity on the majorization lattice
Provides a unified structural understanding of supermodularity and subadditivity for entropic functionals on the majorization lattice, benefiting information theory and convex analysis.
The paper establishes two majorization relations that imply supermodularity and subadditivity of sum-concave functions on the majorization lattice, and proves these properties for Tsallis and Rényi entropies, including strict versions.
We establish two structural majorization relations, which we call precursors, underlying the properties of supermodularity and subadditivity on the lattice induced by majorization. These are precursors in that they immediately imply that all sums of concave functions, which we dub sum-concave functions, are supermodular and subadditive on the majorization lattice. Using these majorization relations, we then show the supermodularity and subadditivity (in the lattice-theoretic sense) of Tsallis entropies (for all $α$) and Rényi entropies (for all $α> 1$), also recovering these properties for the Shannon entropy in the process. We further strengthen these inequalities, showing that: (i) all these entropic functionals are strictly subadditive on the majorization lattice; (ii) Tsallis entropies (and therefore the Shannon entropy as well) are strictly supermodular on the majorization lattice.