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

Zakaria Derbazi

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.

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

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.

Paper Structure

This paper contains 13 sections, 32 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalised Unimodality of Sequences
  3. The sufficient and necessary condition for $m$-modality
  4. Preservers and Reversers of Unimodality
  5. The impact of sign-regularity on unimodal sequences
  6. Preserving Unimodality for Differences, Ratios and Sums
  7. Preservers and Reversers of $m$-modality/$m$-convexity
  8. The relation between higher-order convexity and $m$-modality
  9. $m$-modality and $m$-convexity of a quotient of sequences
  10. Applications and Examples
  11. Domination Polynomials
  12. Numerical examples
  13. Conclusion

Key Result

Theorem 2.8

Let $u$ be an infinite real sequence on $I \subseteq \mathbb{N}$. $u$ is modal of order $m \in \mathbb{N}$ iff $u$ is $M$-regular for some positive. Moreover, if such $M$ exists then $2 \le 2m \le M$.

Figures (1)

  • Figure 1: $m$-modal sequences and their additive inverses.

Theorems & Definitions (78)

  • Definition 2.1: Number of sign changes
  • Definition 2.2: $m$-modal function
  • Definition 2.3: $m$-modal sequence
  • Remark 2.4
  • Definition 2.5: Unimodal $m$-partition and $m$-decomposition of a sequence
  • Definition 2.6: Max-alignment and mode-alignment
  • Definition 2.7: $M$-Regular sequence
  • Theorem 2.8
  • proof
  • Corollary 2.9
  • ...and 68 more