Majorization precursors to supermodularity and subadditivity on the majorization lattice

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 $\alpha$) and R\'enyi entropies (for all $\alpha > 1$), 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.