The Monotonicity and Log-Behaviour 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. On the one hand, a criterion for the monotonicity of the function x f (x) is given, which is a continuous analog for one result of Wang and Zhu. On the other hand, the log-behavior of the functionsand Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. As consequences, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and { n √ z n } for some combinatorial sequences (including the *