Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications

In the present paper, we establish the complete monotonicity of two functions involving divided differences of the digamma function $\psi$ and the trigamma function $\psi'$. Applying these monotonicity, we provide an alternative proof for the monotonicity and convexity of a function derived from bounding the ratio of two gamma functions, procure the logarithmically completely monotonic property of a function involving the ratio of two gamma functions, and obtain new bounds for the ratio of two gamma functions.