Loewner matrices of matrix convex and monotone functions

Abstract

The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$. Similar characterizations are also obtained for matrix monotone functions on a finite interval $(a,b)$.