  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 $(a,b)$.
