Inequalities between ∥ f( A + B)∥ and ∥ f( A) + f( B)∥

Abstract

The conjecture posed by Aujla and Silva [J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217–233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices A j and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ⦀ f( A 1) + f( A 2) + ⋯ + f( A m )⦀ ⩽ ⦀ f( A 1 + A 2 + ⋯ + A m )⦀ holds for any unitarily invariant norm ⦀ · ⦀. It is also proved that ⦀ f( A 1) + f( A 2) + ⋯ + f( A m )⦀ ⩾ f(⦀ A 1 + A 2 + ⋯ + A m ⦀), where f is a non-negative concave function on [0, ∞) and ⦀ · ⦀ is normalized.