On norm sub-additivity and super-additivity inequalities for concave and convex functions

TL;DR

This paper surveys matrix inequalities related to sub-additivity and super-additivity of concave and convex functions, introduces dominated majorization for Hermitian matrices, and provides new proofs and counterexamples for related inequalities.

Contribution

It introduces the concept of dominated majorization between Hermitian matrices and uses it to derive new elementary proofs of known inequalities.

Findings

Dominated majorization reduces to linear majorization under certain conditions.

New elementary proofs for sub-additivity and super-additivity inequalities.

Counterexamples to conjectures extending Ando's inequality to broader classes.

Abstract

Sub-additive and super-additive inequalities for concave and convex functions have been generalized to the case of matrices by several authors over a period of time. These lead to some interesting inequalities for matrices, which in some cases coincide with, and in other cases are at variance with the corresponding inequalities for real numbers. We survey some of these matrix inequalities and do further investigations into these. We introduce the novel notion of dominated majorization between the spectra of two Hermitian matrices $B$ and $C$, dominated by a third Hermitian matrix $A$. Based on an explicit formula for the gradient of the sum of the $k$ largest eigenvalues of a Hermitian matrix, we show that under certain conditions dominated majorization reduces to a linear majorization-like relation between the diagonal elements of $B$ and $C$ in a certain basis. We use this notion as a tool to give new, elementary proofs for the sub-additivity inequality for non-negative concave functions first proved by Bourin and Uchiyama and the corresponding super-additivity inequality for non-negative convex functions first proven by Kosem. Finally, we present counterexamples to some conjectures that Ando's inequality for operator convex functions could more generally hold, e.g.\ for ordinary convex, non-negative functions.