A simple proof of higher order Tur\'{a}n inequalities for Boros-Moll sequences

TL;DR

This paper presents a simplified proof of higher order Turán inequalities for Boros-Moll sequences using a criterion based on checking four inequalities, leveraging bounds from previous studies, and establishing new bounds that imply log-concavity.

Contribution

It introduces a novel, simpler proof method for higher order Turán inequalities for Boros-Moll sequences based on a criterion requiring only four inequalities, utilizing existing bounds and deriving new ones.

Findings

Established a new proof approach for Turán inequalities

Derived a lower bound implying log-concavity of the sequences

Proposed a sharper lower bound for future inequality studies

Abstract

Recently, the higher order Tur\'{a}n inequalities for the Boros-Moll sequences $\{d_\ell(m)\}_{\ell=0}^m$ were obtained by Guo. In this paper, we show a different approach to this result. Our proof is based on a criterion derived by Hou and Li, which need only checking four simple inequalities related to sufficiently sharp bounds for $d_\ell(m)^2/(d_{\ell-1}(m)d_{\ell+1}(m))$. In order to do so, we adopt the upper bound given by Chen and Gu in studying the reverse ultra log-concavity of Boros-Moll polynomials, and establish a desired lower bound for $d_\ell(m)^2/(d_{\ell-1}(m)d_{\ell+1}(m))$ which also implies the log-concavity of $\{\ell! d_\ell(m)\}_{\ell=0}^m$ for $m\geq 2$. We also show a sharper lower bound for $d_\ell(m)^2/(d_{\ell-1}(m)d_{\ell+1}(m))$ which may be available for some deep results on inequalities of Boros-Moll sequences.