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

摘要

Sub-additive and super-additive inequalities for concave and convex functionshave been generalized to the case of matrices by several authors over a periodof time. These lead to some interesting inequalities for matrices, which insome cases coincide with, and in other cases are at variance with thecorresponding inequalities for real numbers. We survey some of these matrixinequalities and do further investigations into these. We introduce the novel notion of dominated majorization between the spectraof 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 certainconditions dominated majorization reduces to a linear majorization-likerelation between the diagonal elements of $B$ and $C$ in a certain basis. Weuse this notion as a tool to give new, elementary proofs for the sub-additivityinequality for non-negative concave functions first proved by Bourin andUchiyama and the corresponding super-additivity inequality for non-negativeconvex functions first proven by Kosem. Finally, we present counterexamples to some conjectures that Ando'sinequality for operator convex functions could more generally hold, e.g.\ forordinary convex, non-negative functions.