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

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