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