Inequalities associated with the Baxter numbers

Abstract

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 \(\{B_{n+1}/B_n\}_{n\geqslant 0}\) and \(\{\hspace{-2.5pt}\sqrt[n]{B_n}\}_{n\geqslant 1}\). Monotonicity of the sequence \(\{\hspace{-2.5pt}\sqrt[n]{B_n}\}_{n\geqslant 1}\) is also obtained. Finally, we prove that the sequences \(\{B_n/n!\}_{n\geqslant 2}\) and \(\{B_{n+1}B_n^{-1}/n!\}_{n\geqslant 2}\) satisfy the higher order Turán inequalities.