Complete monotonicity of some functions involving polygamma functions

摘要

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