Complete monotonicity of some functions involving polygamma functions

In the present paper, we establish necessary and sufficient conditions for the functions xα|ψ(i)(x+β)|xα|ψ(i)(x+β)| and α|ψ(i)(x+β)|−x|ψ(i+1)(x+β)|α|ψ(i)(x+β)|−x|ψ(i+1)(x+β)| respectively to be monotonic and completely monotonic on (0,∞)(0,∞), where i∈Ni∈N, α>0α>0 and β≥0β≥0 are scalars, and ψ(i)(x)ψ(i)(x) are polygamma functions.