Loewner matrices and operator convexity

Abstract

Let f be a function from R+ into itself. A classic theorem of K. Löwner says that f is operator monotone if and only if all matrices of the form [f(p_i)-f(p_j)/p_i-p_j] are positive semidefinite. We show that f is operator convex if and only if all such matrices are conditionally negative definite and that f (t) = t g(t) for some operator convex function g if and only if these matrices are conditionally positive definite. Elementary proofs are given for the most interesting special cases f (t) = t ^r , and f (t) = t log t. Several consequences are derived.