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.