Properties arising from Laguerre-Pólya class for the Boros-Moll numbers

The Boros-Moll numbers $d_i\left(m\right)$ arise from a quartic integral studied by Boros and Moll. For fixed m, the sequence ${\left\{d_i\right(m\left)\right\}}_{0\le i\le m}$ has been proven to satisfy the Turán inequality, the higher order Turán inequality and 3-log-concavity which are originated from the Laguerre-Pólya class. In this paper, we give sharper bounds for both $d_i(m+1)/d_i\left(m\right)$ and $d_i{\left(m\right)}^2/\left(d_{i-1}\right(m\left)d_{i+1}\right(m\left)\right)$. Applying these bounds, we prove a series of results on the log-behavior, the higher order Turán inequality and the Laguerre inequalities of $d_i\left(m\right)$ for fixed i. In our proofs, we use Mathematica as an auxiliary tool to prove inequalities involving several variables. Moreover, we propose a series of open problems.