  We obtain so far unproved properties of a ratio involving a class of Hermite
and parabolic cylinder functions. Those ratios are shown to be strictly
decreasing and bounded by universal constants. Differently to usual analytic
approaches, we employ simple purely probabilistic arguments to derive our
results. In particular, we exploit the relation between Hermite and parabolic
cylinder functions and the eigenfunctions of the infinitesimal generator of the
Ornstein-Uhlenbeck process. As a byproduct, we obtain Tur\'an type
inequalities.
