Some completely monotonic functions involving the gamma and polygamma functions

Abstract

The function [Γ(x + 1)]^1/x(1 + ^1/x)^x/x is strictly logarithmically completely monotonic in (0, ∞). The function ψ″ (x + 2) + (1 + x²)/x²(1 + x)² is strictly completely monotonic in (0, ∞).