Monotone matrix functions of successive orders

This paper extends a result obtained by Wigner and von Neumann. We prove that a non-constant real-valued function, $f(x)$, in $C^3(I)$ where $I$ is an interval of the real line, is a monotone matrix function of order $n+1$ on $I$if and only if a related, modified function $g_{x_0}(x)$ is a monotone matrix function of order $n$ for every value of $x_0$ in $I$, assuming that $f'$ is strictly positive on $I$.