On the Convexity of Functions

Let A, B, and X be bounded linear operators on a separable Hilbert space such that A, B are positive, X >= yl, for some positive real number gamma, and alpha is an element of [0,1]. Among other results, it is shown that if f (t) is an increasing function on [0, infinity) with f (0) = 0 such that f (root t) is convex, then gamma parallel to vertical bar f(alpha A + (1 - alpha) B) + f(beta vertical bar A-B1)parallel to vertical bar <= alpha f (A) X + (1 -alpha) Xf (B) parallel to vertical bar for every unitarily invariant norm, where beta = min (alpha, 1 alpha). Applications of our results are given.