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

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