Monotonicity results for the polygamma functions

Abstract

In this paper, the monotonic results of the functions x c |ψ (n) (x + β)| and xψ (n+1) (x + β)/ψ (n) (x + β) are established. Several by-products are obtained. Moreover, we prove that the function dψ (ln x) dx is completely monotonic. (2000): 33B15, 26A48. Mathematics subject classification