Unimodality preservation by ratios of functional series and integral transforms

TL;DR

This paper establishes conditions under which ratios of functional series and integral transforms preserve unimodality, extending classical results and applying to many important transforms and special functions.

Contribution

It generalizes unimodality preservation criteria to ratios of broad classes of series and transforms, using the concept of sign regularity.

Findings

Many classical transforms satisfy the sufficiency conditions

Unimodality is preserved under these ratios for various special functions

Provides a unified framework for unimodality preservation in functional series

Abstract

An elementary, but very useful lemma due to Biernacki and Krzy\.{z} (1955) asserts that the ratio of two power series inherits monotonicity from that of the sequence of ratios of their respective coefficients. Over the last two decades it has been realized that, under some additional assumptions, similar claims hold for more general series ratios as well as for unimodality in place of monotonicity. This paper continues this line of research: we consider ratios of general functional series and integral transforms and furnish natural sufficiency conditions for preservation of unimodality by such ratios. Numerous series and integral transforms appearing in applications satisfy our sufficiency conditions, including Dirichlet, factorial and inverse factorial series, Laplace, Mellin and generalized Stieltjes transforms, among many others. Finally, we illustrate our general results by exhibiting certain statements on monotonicity patterns for ratios of some special functions. The key role in our considerations is played by the notion of sign regularity.