The extended reverse ultra log-concavity of transposed Boros-Moll sequences

The Boros-Moll sequences $\{d_\ell(m)\}_{\ell=0}^m$ arise in the study of evaluation of a quartic integral. After the infinite log-concavity conjecture of the sequence $\{d_\ell(m)\}_{\ell=0}^m$ was proposed by Boros and Moll, a lot of interesting inequalities on $d_\ell(m)$ were obtained, although the conjecture is still open. Since $d_\ell(m)$ has two parameters, it is natural to consider the properties for the sequences $\{d_\ell(m)\}_{m\ge \ell}$, which are called the \emph{transposed Boros-Moll sequences} here. In this paper, we mainly prove the extended reverse ultra log-concavity of the transposed Boros-Moll sequences $\{d_\ell(m)\}_{m\ge \ell}$, and hence give an upper bound for the ratio ${d_\ell^2(m)}/{(d_\ell(m-1)d_\ell(m+1))}$. A lower bound for this ratio is also established which implies a result stronger than the log-concavity of the sequences $\{d_\ell(m)\}_{m\ge \ell}$. As a consequence, we also show that the transposed Boros-Moll sequences possess a stronger log-concave property than the Boros-Moll sequences do. At last, we propose some conjectures on the Boros-Moll sequences and their transposes.