Some Complete Monotonicity Properties Of The Gamma And Psi Functions With Applications

It has been more than 200 years since Euler started to systematically study Gamma function up to now. As a special transcendental function, Gamma function has many good properties and plays an important role in the branch of mathematics, physics, engineering etc. The Psi function, the logarithmic derivative of the Gamma function, is another special function that we are to study in the paper. In recent years, there exists a very extensive literature on these two functions. In particular, inequalities for these functions have been published. It is well known that many inequalities derive in monotonicity and complete monotonicity properties for these functions.This paper contains four chapters:Chapter 1 mainly introduces some definitions and some well-known or known theorems. Firstly, we introduce the definitions of Gamma function, Psi function, completely monotonic function etc. and give some well-known formulas such as functional equation, complement formula, Weierstrass formula etc. Subsequently, we introduce the history and the present development of Gamma and Psi function.In chapter 2, the logarithmically completely monotonic results of the functions related to the Gamma function are obtained. These theorems generalize and extend some results of H.Vogt, Merkle and Ch.-P. Chen. Then we obtain the completely monotonic property of several functions that contain Psi function by the relations to Gamma function and Psi function and the properties of completely monotonic function. Lastly, we extend some known conclusions, at the same time, we obtain some inequalities of Gamma function and Psi function.Chapter 3 mainly researches the completely monotonic property related to the Psi function and some applications. These theorems generalize some results of S.-L.Qiu and M.Vurinen .In chapter 4, we introduce the Log-convex property of completely monotonic function and other monotonic functions.