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 $F(x)=\sqrt[x]{ \frac{ \Gamma(ax+b+1)}{ \Gamma(c x+d+1)\Gamma(e x+f+1)}}$ 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 St\u{ a}nic\u{ a}. (3). By researching logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequence $\{ \frac{ (n_{ 0}+ia)!}{ (k_0+ib)!(\overline{ k_0}+i\overline{ b})!}\}_{ i\geq0}$ is proved. This generalizes two results of Chen { \it et al.} that both the Catalan numbers $\frac{ 1}{ n+1}\binom{ 2n}{ n}$ and central binomial coefficients $\binom{ 2n}{ n}$ are infinitely log-monotonic and strengths one result of Su and Wang that $\binom{ dn}{ \delta n}$ 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 Tepedelenlio\v{ g}lu, Qi and Li.
Categories: math.CO