2008/01/23 by P. J. Rippon, Rippon, P. J., Gwyneth M. Stallard +2 · 1 citation
Mathematics · #30D05 #37F10 #Analytic Number Theory Research #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #math.CV #math.DS #msc:30D05 #msc:37F10
paper · pdf · doi:10.48550/arxiv.0801.3610
arxiv created 2008/01/23 · openalex publication_date 2008/01/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a form of the cos πρ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen's condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components, also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.