On Preserving or Reversing Higher-Order Unimodality and Convexity by Sign-Regular Kernels

TL;DR

This paper characterizes when sign-regular kernels preserve or reverse unimodality and convexity in sequences, extending classical results and providing conditions for higher-order properties.

Contribution

It establishes necessary and sufficient conditions for sign-regular kernels to preserve or reverse higher-order unimodality and convexity, expanding Karlin's classical results.

Findings

Preservation occurs only if kernels are totally positive of order three or their inverse is totally negative of order three.

Reversal occurs if kernels are totally negative or their inverse is totally positive of order three.

Results extend to higher-order convexity and multimodal sequences.

Abstract

This work investigates preserving and reversing unimodality and convexity properties for sequences under transformations defined by sign-regular kernels. It is shown that these transformations only preserve these properties if the kernels are totally positive of order three or their additive inverse is totally negative of order three. In contrast, these transformations reverse these properties if the underlying kernel is totally negative or if its additive inverse is a totally positive kernel, both of order three. Furthermore, these results are extended to higher-order convex and multimodal sequences. These findings, which expand upon Karlin's earlier results on convexity, form the basis for deriving sufficient conditions for the preservation or reversal of higher-order convexity or generalised unimodality of a quotient of sequences, where both the numerator and denominator are transformations by the same sign-regular kernel.