Complete monotonicity of two functions involving the tri-and tetra-gamma functions

Abstract

The psi function ψ(x) is defined by ψ(x) = Γ′(x)/Γ(x) and ψ (i) (x), for i ∈ ℕ, denote the polygamma functions, where Γ(x) is the gamma function. In this paper, we prove that the functions \( [\psi '(x)]^2 + \psi ''(x) - \frac{{x^2 + 12}} {{12x^4 (x + 1)^2 }} \)and \( \frac{{x + 12}} {{12x^4 (x + 1)}} - \{ [\psi '(x)]^2 + \psi ''(x)\} \) are completely monotonic on (0,∞).