Abstract
In this paper, we study the completely monotonic property of two functions involving the function xe002;(x) = [psi(x)]2 + psi(x) and deduce the double inequality x2 + 3x + 3 625x2 + 2275x + 5043 3x4(2x + 1)2 < xe002;(x) <3x4(50x + 41)2 , x > 0 which improve some recent results, where psi(x) is the logarithmic derivative of the Gamma function. Also, we deduce the completely monotonic degree of a function involving psi(x).