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

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 Aj and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ⦀f(A1) + f(A2) + ⋯ + f(Am)⦀ ⩽ ⦀ f(A1 + A2 + ⋯ + Am)⦀ holds for any unitarily invariant norm ⦀ · ⦀. It is also proved that ⦀f(A1) + f(A2) + ⋯ + f(Am)⦀ ⩾ f(⦀A1 + A2 + ⋯ + Am⦀), where f is a non-negative concave function on [0, ∞) and ⦀ · ⦀ is normalized.