Completely monotonic degree of a function involving the tri- and tetra-gamma functions

Abstract: Let $\psi(x)$ be the di-gamma function, the logarithmic derivative of the classical Euler's gamma function $\Gamma(x)$. In the paper, the author shows that the completely monotonic degree of the function $[\psi'(x)]^2+\psi''(x)$ is $4$, surveys the history and motivation of the topic, supplies a proof for the claim that a function $f(x)$ is strongly completely monotonic if and only if the function $xf(x)$ is completely monotonic, conjectures the completely monotonic degree of a function involving $[\psi'(x)]^2+\psi''(x)$, presents the logarithmic concavity and monotonicity of an elementary function, and poses an open problem on convolution of logarithmically concave functions.