The Complete Monotonicity Of Some Special Functions Associated With The Gamma Function

The specially and completely monotonic functions what associated with the Gamma function——especially Psi function and digamma function——have comprehensive applications in different branches of mathematical sciences. For example, they have some important role in combinatorics, numerical and asymptotic analysis, physic, potential theory, and probability theory. In this paper, the complete monotonicity of special function were deeply study. In the article, firstly, the concept and nature of the completely monotonic function and logarithmically completely monotonic function were introduced. Secondly, the completely monotonicity or logarithmically completely monotonicity of some special functions related to gamma function were proved. Thus, some new important inequalities were obtained. Lastly, these new inequality were compared with the existing conclusions.