The Completely Monotonic Functions And Logarithmically Completely Monotonic Functions Involving The Gamma Function

The completely monotonic property and logarithmically completely monotonic property are two important properties of the gamma function and functions related to the gamma function, and they play a important role in many fields, such as number theory, probability theory, differential equation, definite integral, riemann's zeta-function and physics and so on. Because of this, More and more scholars devote themselves to the studies of the (logarithmically) completely monotonic theory involving the gamma function. And they acquire great results. In this paper, we mainly prove some logarithmically completely mono-tonic functions involving the gamma function and inequalities. First, using the monotonicity theory, the series expansions and the integral representations, this paper mainly study logarithmically completely monotonic properties of functions and provides the sufficient condition that makes the function f_?(x) logarithmically completely monotonic on(0,?) as well as attains a inequality which is more accurate than the original conclusion and a two-side inequality; Second, using the monotonicity theory, the integral representations and the induction, this paper mainly researches completely monotonic properties of functions F_?(x)=??(x)+(?/x)-(1/(x+1)),f_?(x)= Inx-?(x)-(?/x) and f(x)=Inx-?(x)+(1/x2) and functions on (0,?). At the same time, two inequalities are attained according to the logarithmically completely monotonic properties of them and are compared.