Completely Monotonic Functions Related To The Gamma Function And Its Applications

The Euler Gamma function, which is widely used in probability and statis-tics, mathematical analysis, physics and other subjects, is the foundation of the special function theory. Functions related to Gamma and Psi function have a very beautiful nature - Complete monotonicity and logarithmically complete monotonicity. This provides a powerful tool for the study of inequalities and, to a great extent, promotes the development of the theory of inequality. This paper studies the complete monotonicity and logarithmically complete monotonicity of functions related to Gamma and Psi function. Based on this, some inequalities is established. The article main content includes:First of all, it is improved the logarithmically complete monotonicity and related inequalities of the function G?,?(x)(where ??R,?> 0 are param-eter) and G?,?(x)/1 which are defined on (0, ?) in this paper. The sufficient condition is expanded by using Taylor series expansion, the series expansion and integral expression of Gamma function and Psi function. Details are as follows. (?) If,?? (0,2/1+2/?3) and ?< min{h2(?), h3{?)}, then G?,?(x) is logarith-mically completely monotonic on (0,?). (?) If ?? [0,2/1-6/?3] and ?> h3(?), then 1/G?,?(x) is logarithmically com-pletely monotonic on (0, ?). (?) On the basis of (?) and (?), some inequalities are established. Based on the research of the special circumstances, a symmetrical and concise two-side inequality, which estimates the division of factorial and ?kn=1 kk, is established.Secondly, this paper constructs a function F?(x) (where the parameter ? ? R) related to the Psi function. It is proved that the function F?(x) is completely monotonic if and only if ?< 2 by using the properties of completely monotonic function and monotonicity of known functions. The way to prove is novel and clever. Based on the complete monotonicity, a new inequality is established.