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