On a conjecture of Chen-Guo-Wang

TL;DR

This paper proves a conjecture on the log-concavity of a function involving the Riemann zeta and Gamma functions, extends it to almost infinite log-monotonicity of certain combinatorial sequences, and confirms related conjectures of Sun.

Contribution

It proves Chen et al.'s conjecture on the log-concavity of a specific function and extends the concept to almost infinite log-monotonicity of Bernoulli and tangent number sequences.

Findings

Proved the log-concavity conjecture for the function θ(x).

Extended the conjecture to almost infinite log-monotonicity of Bernoulli and tangent sequences.

Established the almost infinite log-monotonicity of specific combinatorial sequences.

Abstract

Towards confirming Sun's conjecture on the strict log-concavity of combinatorial sequence involving the n$th$ Bernoulli number, Chen, Guo and Wang proposed a conjecture about the log-concavity of the function $\theta(x)=\sqrt[x]{2\zeta(x)\Gamma(x+1)}$ for $x\in (6,\infty)$, where $\zeta(x)$ is the Riemann zeta function and $\Gamma(x)$ is the Gamma function. In this paper, we first prove this conjecture along the spirit of Zhu's previous work. Second, we extend Chen et al.'s conjecture in the sense of almost infinite log-monotonicity of combinatorial sequences, which was also introduced by Chen et al. Furthermore, by using an analogue criterion to the one of Chen, Guo and Wang, we deduce the almost infinite log-monotonicity of the sequences $\frac{1}{\sqrt[n]{|B_{2n}|}}$, $T_n$ and $\frac{1}{\sqrt[n]{T_n}}$, where $B_{2n}$ and $T_{n}$ are the $2n$th Bernoulli number and the $n$th tangent number, respectively. These results can be seen as extensions of some solved conjectures of Sun.