Universal bounds and monotonicity properties of ratios of Hermite and parabolic cylinder functions

Abstract

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 Turán type inequalities.

Uniform (very) sharp bounds for ratios of parabolic cylinder functions

“…The previous analysis clearly suggests that should be an increasing function of x . However, from the previous inequalities alone it does not seem possible to prove this result, which is known to be true and has been proved by indirect arguments 7 . Here, we give a direct proof of this result and, as a by‐product of this analysis, we obtain the tightest available bounds for the ratios and double ratios of the PCFs.…”

“…We will not only prove the monotonicity property described in Ref. [7], but we will obtain from this analysis new and very sharp bounds for ratios of the PCFs.…”

“…Remark The central inequalities reappeared in Ref. [7], but they were already proved in Ref. [8].…”

“…Despite their relevance, little is known regarding simple functional uniform approximations and bounds, 7,8 which is in contrast to the vast information available on other functions like, for instance, modified Bessel functions (see Refs. [9][10][11][12] and references cited therein).…”

“…[9][10][11][12] and references cited therein). Recently, one new monotonicity property was discovered for ratios of the PCFs from stochastic considerations 7 that had not been proved by purely analytical methods before; this fact illustrates that there is a clear deficit in the amount of analytical information on these functions.…”