Monotonicity and log-behavior of some functions related to the Euler Gamma function

Abstract

The aim of this paper is to develop analytic techniques to deal with certain monotonicity of combinatorial sequences. (1) A criterion for the monotonicity of the function $\sqrt[x]{f(x)}$ is given, which is a continuous analog for one result of Wang and Zhu. (2) The log-behavior of the functions $\theta (x)=\sqrt[x]{2\zeta (x)\Gamma (x+1)}$ and $F(x)=\sqrt[x]{\frac{\Gamma (ax+b+1)}{\Gamma (cx+d+1)\Gamma (ex+f+1)}}$ is considered, where $\zeta (x)$ and $\Gamma (x)$ are the Riemann zeta function and the Euler Gamma function, respectively. As consequences, the strict log-concavities of the function $\theta (x)$ (a conjecture of Chen {\it et al.}) and $\{\sqrt[n]{z_n}\}$ for some combinatorial sequences (including the Bernoulli numbers, the Tangent numbers, the Catalan numbers, the Fuss-Catalan numbers and some Binomial coefficients) are demonstrated. In particular, this contains some results of Chen {\it et al.}, Luca and Stănică. (3). By researching logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequence $\{\frac{(n_0+ia)!}{(k_0+ib)!(\overline{k_0}+i\overline{b})!}{\}}_{i\ge 0}$ is proved. This generalizes two results of Chen {\it et al.} that both the Catalan numbers $\frac{1}{n+1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$ and central binomial coefficients $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$ are infinitely log-monotonic and strengths one result of Su and Wang that $\left(\genfrac{}{}{0ex}{}{dn}{\delta n}\right)$ is log-convex in $n$. (4) The asymptotically infinite log-monotonicity of derangement numbers is showed. (5)The logarithmically complete monotonicity of functions $1/\sqrt[x]{a\zeta (x+b)\Gamma (x+c)}$ and $\sqrt[x]{\rho {\prod }_{i=1}^n\frac{\Gamma (x+a_i)}{\Gamma (x+b_i)}}$ is also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, Qi and Li.

to appear in Proceedings of the Edinburgh Mathematical Society (2015)