A Simple Proof of Higher Order Turán Inequalities for Boros–Moll Sequences

Recently, the higher order Turán inequalities for the Boros–Moll sequences {dℓ(m)}ℓ=0m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d_\ell (m)\}_{\ell =0}^m$$\end{document} 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 dℓ(m)2/(dℓ-1(m)dℓ+1(m))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\ell (m)^2/(d_{\ell -1}(m)d_{\ell +1}(m))$$\end{document}. 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 dℓ(m)2/(dℓ-1(m)dℓ+1(m))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\ell (m)^2/(d_{\ell -1}(m)d_{\ell +1}(m))$$\end{document} which also implies the log-concavity of {ℓ!dℓ(m)}ℓ=0m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\ell ! d_\ell (m)\}_{\ell =0}^m$$\end{document} for m≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 2$$\end{document}.