A note on definition of matrix convex functions

Abstract

We prove that a real-valued function f defined on an interval S in R is matrix convex if and only if for any natural k, for all families of positive operators { Ai }i = 1 k in a finite-dimensional Hilbert space, such that ∑i = 1 k Ai = 1, and arbitrary numbers xi ∈ S, the inequality{A formula is presented}holds true. © 2006 Elsevier Inc. All rights reserved.