A class of completely monotonic functions involving the gamma and polygamma functions

In the paper the authors show that neither the function In Gamma(x) - (x - 1/2) In x + x - 1/2In x+x - 1/2In(2 pi) -a(1)[-1)(i+1)psi((l))(x+b(l))] nor its negative is completely monotonic on (0, infinity), where a i >= 0 and b i = 0 are real numbers, i >= 2, Gamma(x) is the classical Euler's gamma function, and psi((i))(x) = [Gamma(x)/Gamma(x)]((i)) are polygamma functions. Moreover, some other results are given in the form of remarks.