The Ramanujan Journal

, Volume 11, Issue 2, pp 225–248

Some classes of completely monotonic functions, II


DOI: 10.1007/s11139-006-6510-5

Cite this article as:
Alzer, H. & Berg, C. Ramanujan J (2006) 11: 225. doi:10.1007/s11139-006-6510-5


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.


Complete monotonicity Gamma digamma and polygamma functions Prime numbers Inequalities 

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.WaldbrölGermany
  2. 2.Department of MathematicsUniversity of CopenhagenDenmark

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