Complete monotonicity of a family of functions involving the tri- and tetra-gamma functions

Abstract

The psi function $ψ(x)$ is defined by $ψ(x)=\frac{Γ'(x)}{Γ(x)}$ and $ψ^{(i)}(x)$ for $i\in\mathbb{N}$ denote polygamma functions, where $Γ(x)$ is the gamma function. In this paper, we prove that the function $$ [ψ'(x)]^2+ψ"(x)-\frac{x^2+λx+12}{12x^4(x+1)^2} $$ is completely monotonic on $(0,\infty)$ if and only if $λ\le0$, and so is its negative if and only if $λ\ge4$. From this, some inequalities are refined and sharpened.