Superadditivity of convex integral transform for positive operators in Hilbert spaces

For a continuous and positive function w (lambda), lambda > 0 and mu a positive measure on (0, infinity) we consider the following convex integral transform C (w, mu) (T) := integral(infinity)(0) w (lambda) T-2 (lambda + T)(-1) d mu(lambda) where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among other that, for all A, B > 0 with BA + AB >= 0, C( w, mu) (A + B) >= C(w, mu) ( A) + C(w, mu) ( B). In particular, we have for r is an element of (0, 1], the power inequality (A + B)(r+1) >= A(r+1) + Br+1 and the logarithmic inequality (A + B) ln (A + B) >= A ln A + B ln B. Some examples for operatormonotone and operator convex functions and integral transforms C (center dot, center dot) related to the exponential and logarithmic functions are also provided.