Two Class of Completely Monotonic Functions Involving Gamma and Polygamma Functions

The function [Γ(x+ 1)]1/x xc ( 1 + 1 x )x is logarithmically completely monotonic in (0,∞) if and only if c ≥ 1 and its reciprocal is logarithmically completely monotonic in (0,∞) if and only if c ≤ 0. The function ψ′′(x) + 2 + (6 + c)x+ (4 + 3c)x2 + (2 + 3c)x3 + cx4 x3(x+ 1)3 is completely monotonic in (0,∞) if and only if c ≥ 1 and its negative is completely monotonic in (0,∞) if and only if c ≤ 0.