Some norm inequalities for completely monotone functions-II

Let A,B be n×n complex positive definite matrices, X any n×n complex matrix and f a completely monotone function on (0,∞). Let m(A,B) denote the smallest among the eigenvalues of A and B. We prove that f(A)X−Xf(B) f′(m(A,B)) AX−XB for all unitarily invariant norms • . This supplements results of Ando, Bhatia, Kittaneh and Kosaki.