Uniqueness of nontrivially complete monotonicity for a class of functions involving polygamma functions

For $m,n\in \mathbb{N}$, let $f_{m,n}(x)=[{\psi }^{(m)}(x){]}^2+{\psi }^{(n)}(x)$ on $(0,\mathrm{\infty })$. In the present paper, we prove using two methods that, among all $f_{m,n}(x)$ for $m,n\in \mathbb{N}$, only $f_{1,2}(x)$ is nontrivially completely monotonic on $(0,\mathrm{\infty })$. Accurately, the functions $f_{1,2}(x)$ and $f_{m,2n-1}(x)$ are completely monotonic on $(0,\mathrm{\infty })$, but the functions $f_{m,2n}(x)$ for $(m,n)\mathrm{\ne }(1,1)$ are not monotonic and does not keep the same sign on $(0,\mathrm{\infty })$.