Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic

The psi function ψ(x) is defined by ψ(x)=Γ′(x)/Γ(x), where Γ(x) is the gamma function. We give necessary and sufficient conditions for the function ψ″(x)+2[ψ′(x+α)] or its negative to be completely monotonic on (−α,∞), where α∈R. We also prove that the function 2[ψ′(x)]+λψ″(x) is completely monotonic on (0,∞) if and only if λ⩽1. As an application of the latter conclusion, the monotonicity and convexity of the function epψ(x+1)−qx with respect to x∈(−1,∞) are thoroughly discussed for p≠0 and q∈R.