Comments on the paper "Universal bounds and monotonicity properties of ratios of Hermite and Parabolic Cylinder functions"

TL;DR

This paper critiques and clarifies previous work on bounds and properties of ratios of Hermite and parabolic cylinder functions, correcting an error and proposing a new conjecture on optimal bounds.

Contribution

It clarifies the validity of earlier results on universal bounds and introduces a conjecture on the tightest possible upper bound for a specific ratio of parabolic cylinder functions.

Findings

Confirmed the correctness of previous universal bounds despite an earlier error.

Identified an error in a referenced proof and provided a fix.

Proposed a new conjecture on the optimal upper bound for a ratio of parabolic cylinder functions.

Abstract

In the abstract of [1] we read: "We obtain so far unproved properties of a ratio involving a classof Hermite and parabolic cylinder functions." However, we explain how some of the main results in that paper were already proved in [2], namely the `universal bounds'. An error in reference [2] was discussed in [1] which does not affect the proof given there for those `universal bounds'; we fix this erratum easily. We end this note proposing a conjecture regarding the best possible upper bound for a certain ratio of parabolic cylinder functions.