A completely monotonic function involving the tri-gamma function and with degree one

In the paper the authors introduce the concept of completely monotonic degrees of nonnegative functions on (0,∞)(0,∞) and proves that a function involving the difference between the tri-gamma function ψ′(x)ψ′(x) and a proper fraction of x   is completely monotonic of degree 1 on (0,∞)(0,∞).