Complete Monotonicity of a Polygamma Function plus the Square of Another Polygamma Function

For m,n ∈ N, let fm,n(x) = ψ (x)+ ˆ ψ(x) ̃2 for x > 0. In the present paper, we prove that f1,2(x) is the only nontrivial completely monotonic function on (0,∞). Accurately, the functions f1,2(x) and fm,2n−1(x) are completely monotonic on (0,∞), but the functions fm,2n(x) for (m,n) 6= (1, 1) are not monotonic and does not keep the same sign on (0,∞).