Loewner Matrices and Operator Convexity

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(pi)−f(pj) pi−pj ] 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) = tg(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, and f(t) = t log t. Several consequences are derived. 2000 Mathematics Subject Classification: 15A48, 47A63, 42A82.