Unimodality Preservation by Ratios of Functional Series and Integral Transforms

An elementary, but very useful lemma due to Biernacki and Krzyż (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.