Some classes of completely monotonic functions, II

Abstract

A function \(f\!:(0,\infty)\rightarrow \mathbf{R}\) is said to be completely monotonic if \((-1)^n f^{(n)}(x)\geq 0\) for all x > 0 and n = 0,1,2,.... In this paper we present several new classes of completely monotonic functions. Our functions have in common that they are defined in terms of the classical gamma, digamma, and polygamma functions. Moreover, we apply one of our monotonicity theorems to prove a new inequality for prime numbers. Some of the given results extend and complement theorems due to Bustoz & Ismail, Clark & Ismail, and other researchers.