Concavity of Monotone Matrix Functions of Finite Order

Let f : (0; 1) ! R be a monotone matrix function of order n for some arbitrary but xed value of n. We show that f is a matrix concave function of order bn=2c and that kf(A) ? f(B)k kf(jA ? Bj)k for all n-by-n positive semideenite matrices A and B, and all unitarily invariant norms k k. Because f is not assumed to be a monotone matrix function of all orders, Loewner's integral representation of functions that are monotone of all orders is not applicable, instead we use the functional characterization of f in proving these results.