2-Log-concavity of the Boros-Moll Polynomials

TL;DR

This paper proves that Boros-Moll polynomials are 2-log-concave for all m ≥ 2, confirming a conjecture and providing new insights into their coefficient sequences and related inequalities.

Contribution

The paper establishes the 2-log-concavity of Boros-Moll polynomials and introduces a new proof of Moll's minimum conjecture, advancing understanding of their coefficient properties.

Findings

Proved 2-log-concavity of Boros-Moll polynomials for all m ≥ 2

Showed the log-concavity of a specific coefficient sequence related to the polynomials

Provided a new proof of Moll's minimum conjecture

Abstract

The Boros-Moll polynomials $P_m(a)$ arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that $P_m(a)$ is 2-log-concave for any $m\geq 2$. Let $d_i(m)$ be the coefficient of $a^i$ in $P_m(a)$. We also show that the sequence $\{i (i+1)(d_i^{\,2}(m)-d_{i-1}(m)d_{i+1}(m))\}_{1\leq i \leq m}$ is log-concave. This leads another proof of Moll's minimum conjecture.