Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In the paper, we prove the completely monotonic (CM) property of some functions involving the function Δ(l)=ψ″(l)+ψ′(l)2 and hence we deduce a new double inequality for it. Additionally, we study the CM degree of some functions involving the function ψ′(l). Our new bounds takes priority over some of the recently published results.