The Reverse Ultra Log-Concavity of the Boros-Moll Polynomials

TL;DR

This paper proves the reverse ultra log-concavity of Boros-Moll polynomials, establishes related inequalities, and discusses the borderline between ultra log-concavity and reverse ultra log-concavity, proposing conjectures for future research.

Contribution

It introduces the reverse ultra log-concavity property for Boros-Moll polynomials and related inequalities, advancing understanding of their coefficient sequences.

Findings

Proved reverse ultra log-concavity of Boros-Moll polynomials.

Established an inequality implying the log-concavity of a related sequence.

Showed Boros-Moll sequences are on the borderline between ultra and reverse ultra log-concavity.

Abstract

We prove the reverse ultra log-concavity of the Boros-Moll polynomials. We further establish an inequality which implies the log-concavity of the sequence $\{i!d_i(m)\}$ for any $m\geq 2$, where $d_i(m)$ are the coefficients of the Boros-Moll polynomials $P_m(a)$. This inequality also leads to the fact that in the asymptotic sense, the Boros-Moll sequences are just on the borderline between ultra log-concavity and reverse ultra log-concavity. We propose two conjectures on the log-concavity and reverse ultra log-concavity of the sequence $\{d_{i-1}(m) d_{i+1}(m)/d_i(m)^2\}$ for $m\geq 2$.