  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 the polygamma functions, where $\Gamma(x)$ is the gamma function. In
this paper we prove that a function involving the difference between
$[\psi'(x)]^2+\psi''(x)$ and a proper fraction of $x$ is completely monotonic
on $(0,\infty)$.
