Gamma函数的相关函数及其q化的完全单调性

Complete monotonicity properties of functions involving the Gamma function and their q-analogues

为研究Gamma函数的相关函数及其q化的完全单调性,首先利用函数的积分表达式以及单调性理论证明函数Fα(x)=ψ′(x)+α/x-1/(x+1),fα(x)=lnx-ψ(x)-α/x以及f(x)=lnx-ψ(x)+1/x^2在(0,∞)上的完全单调性;其次基于q化的性质,利用函数的级数展开、单调性理论以及归纳法讨论上述相应函数的q化的完全单调性,并利用该性质得到2个不等式.

A investigation to derive the complete monotonicities of functions involving the Gamma function and their q-analogues is presented.First of all,using the integral representations and the monotonicity theory,it is proved that the complete monotonicities of functions Fα(x)=ψ′(x)+α/x-1/(x+1),fα(x)=lnx-ψ(x)-α/xand f(x)=lnx-ψ(x)+1/x^2 on(0,∞);Second,based on the properties of q-analogues,the complete monotonicities of q-analogues about these three functions are respectively discussed by the series expansion of functions,the monotonicity theory and induction,and two inequalities are obtained because of the property.