Inequalities associated with the Baxter numbers

The Baxter numbers (B_n) enumerate a lot of algebraic and combinatorial objects such as the bases for subalgebras of the Malvenuto-Reutenauer Hopf algebra and the pairs of twin binary trees on (n) nodes. The Turán inequalities and higher order Turán inequalities are related to the Laguerre-Pólya ((mathcal{L})-(mathcal{P})) class of real entire functions, and the (mathcal{L})-(mathcal{P}) class has a close relation with the Riemann hypothesis. The Turán type inequalities have received much attention. In this paper, we are mainly concerned with Turán type inequalities, or more precisely, the log-behavior, and the higher order Turán inequalities associated with the Baxter numbers. We prove the Turán inequalities (or equivalently, the log-concavity) of the sequences ({B_{n+1}/B_n}_{ngeqslant 0}) and ({hspace{-2.5pt}sqrt[n]{B_n}}_{ngeqslant 1}). Monotonicity of the sequence ({hspace{-2.5pt}sqrt[n]{B_n}}_{ngeqslant 1}) is also obtained. Finally, we prove that the sequences ({B_n/n!}_{ngeqslant 2}) and ({B_{n+1}B_n^{-1}/n!}_{ngeqslant 2}) satisfy the higher order Turán inequalities

A simple proof of higher order Tur\'{a}n inequalities for Boros-Moll sequences

Recently, the higher order Tur\'{a}n inequalities for the Boros-Moll sequences $\{d_\ell(m)\}_{\ell=0}^m$ 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 $d_\ell(m)^2/(d_{\ell-1}(m)d_{\ell+1}(m))$. 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 $d_\ell(m)^2/(d_{\ell-1}(m)d_{\ell+1}(m))$ which also implies the log-concavity of $\{\ell! d_\ell(m)\}_{\ell=0}^m$ for $m\geq 2$. We also show a sharper lower bound for $d_\ell(m)^2/(d_{\ell-1}(m)d_{\ell+1}(m))$ which may be available for some deep results on inequalities of Boros-Moll sequences.Comment: 12 page