Glasnik Matematicki, Vol. 58, No. 1 (2023), 1-16. \( \)

INEQUALITIES ASSOCIATED WITH THE BAXTER NUMBERS

James Jing Yu Zhao

School of Accounting, Guangzhou College of Technology and Business, Foshan, Guangdong 528138, China
e-mail:zhao@gzgs.edu.cn

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.

2020 Mathematics Subject Classification.   05A20, 11B83

Key words and phrases.   Log-concavity, log-convexity, log-balancedness, higher order Turán inequalities, Baxter numbers

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.58.1.01

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