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 x n complex matrices A(j) and for any non-negative convex function f on [0, infinity) with f (0) = 0 the inequality vertical bar vertical bar vertical bar f(A(1)) + f(A(2)) + center dot center dot center dot + f(A(m))vertical bar vertical bar vertical bar <= vertical bar vertical bar vertical bar f(A(1) + A(2) + center dot center dot center dot + A(m))vertical bar vertical bar vertical bar holds for any unitarily invariant norm vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar. It is also proved that vertical bar vertical bar vertical bar f(A(1)) + f(A(2)) + center dot center dot center dot + f(A(m))vertical bar vertical bar vertical bar >= f(vertical bar vertical bar vertical bar A(1) + A(2) + center dot center dot center dot + A(m)vertical bar vertical bar vertical bar), where f is a non-negative concave function on [0, infinity) and vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar is normalized. (c) 2006 Elsevier Inc. All rights reserved.