The Complete Monotonicity And Inequality About The Ratio Of Gamma Function And Q-psi Function

In this paper,A series of functions containing the ratio of Gamma function were constructed,and their completely monotonicity was proved.we also ob-tained strict bidirectional inequality under general form.Then we proved the the complete monotonicity for functions involving q-digamma functions,some appli-cations of these results give inequalities containing q-digamma and q-trigamma functions.The first part of the paper mainly studies the bidirectional inequality and logarithmically completely monotonicity about the ratio of gamma func-tion.In 2014,affected by the inequality contains the ratio of wallis,C.Mortiei,V.G.Cristea and dawei Lu studied the product about and Constructed and proved the monotonicity completely contains P1 and P2.and established some strict in-equalities about them.This paper studies the ratio of gamma function about Ti=Γ(n+i/4)/Γ(n+1)Γ(i/4),i=1,2,3.Constructed and proved the monotonicity completelycontains Ti,and established some inequalities about Ti.By introducing the error sequence,the error range of bidirectional inequality was improved,and general-ized the type of the ratio of Gamma function to general form,we also obtained strict bidirectional inequality about n,n≥1,n∈N,i=1,2,3...k-1.And proved a new form of inequality contains Wallis ratio.The second part of the paper mainly proved the complete monotonicity about q-digamma function.By introducing the classical q-theory,to simulate the function contains digamma function.using the nature of the series and q-simulate function,the complete monotonicity for functions involving q-digamma functions are proved,some applications of these results give inequalities contain-ing q-digamma and q-trigamma functions.