  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.
