A Simple Proof of Higher Order Turan Inequalities for Boros-Moll Sequences

Recently, the higher order Turan inequalities for the Boros-Moll sequences {d(l)(m)}(m)(l=0) 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(l)(m)(2)/(d(l-1)(m)d(l+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(l)(m)(2)/(d(l-1)(m)d(l+1)(m)) which also implies the log-concavity of {l!d(l)(m)}(m)(l=0) for m = 2.