Completely monotonic degree of a function involving trigamma and tetragamma functions

Let $\psi(x)$ be the digamma function. In the paper, the author reviews backgrounds and motivations to compute complete monotonic degree of the function $\Psi(x)=[\psi'(x)]^2+\psi''(x)$ with respect to $x\in(0,\infty)$, confirms that completely monotonic degree of the function $\Psi(x)$ is $4$, finds a relation between strongly completely monotonic functions and completely monotonic degrees, provides a proof for the relation between strongly completely monotonic functions and completely monotonic degrees, proves a property of logarithmically concave functions, and poses two open problems on lower bound for convolution of logarithmically concave functions and on completely monotonic degree of a function involving $\Psi(x)$.