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Glasnik matematički, Vol. 58 No. 1, 2023.

Original scientific paper

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

Inequalities associated with the Baxter numbers

James Jing Yu Zhao orcid id orcid.org/0000-0002-3001-6738 ; School of Accounting, Guangzhou College of Technology and Business, Foshan, Guangdong 528138, China

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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.

Keywords

Log-concavity, log-convexity, log-balancedness, higher order Turán inequalities, Baxter numbers

Hrčak ID:

304386

URI

https://hrcak.srce.hr/304386

Publication date:

20.6.2023.

Visits: 390 *

Publication date: 30 June 2023

Volume: Vol 58

Issue: Svezak 1

Pages: 1-16

DOI: 10.3336/gm.58.1.01

Article Information (continued)

Categories:

Subject: Izvorni znanstveni članak

Keywords:

Keyword: Log-concavity, log-convexity, log-balancedness, higher order Turán inequalities, Baxter numbers

Inequalities associated with the Baxter numbers

James Jing Yu Zhao[1]

Email: zhao@gzgs.edu.cn

 

[1] School of Accounting, Guangzhou College of Technology and Business, Foshan, Guangdong 528138, China

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.

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