Complete monotonicity of some functions involving polygamma functions

In the present paper, we establish necessary and sufficient conditions for the functions x α | 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.