A simple proof of higher order Turán inequalities for Boros-Moll sequences (Q6040454)

This paper is a different approach to a recent result concerning higher-order Turán inequalities for the Boros-Moll sequence \(\{d_l(m)\}_{l=0}^m\) obtained by \textit{J. J. F. Guo} [J. Number Theory 225, 294--309 (2021; Zbl 1465.05017)]. Here, \(d_l\) is the coefficient of \(x^l\) in the Boros-Moll polynomials \[ P_m(x)=\sum_{j,k}\binom{2m+1}{2j}\binom{m-j}{k}\binom{2k+2j}{k+j}\frac{(x+1)^j(x-1)^k}{2^{3(k+j)}}. \] These polynomials arise in the study of a quartic integral \[ \int_0^\infty \frac{dt}{{(t^4+2xt^2+1)}^{m+1}}=\frac{\pi}{2^{m+3/2}(x+1)^{m+1/2}}P_m(x). \] The paper discusses, among other things, a sharper bound for \(\frac{d_l(m+1)}{d_l(m)}\), and one for \(\frac{d_l(m)^2}{d_{l-1}(m)d_{l+1}(m)}\).