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Functions of small growth with no unbounded Fatou components

Rippon, P. J. and Stallard, G. M. (2009). Functions of small growth with no unbounded Fatou components. Journal d’Analyse Mathematique, 108(1) pp. 61–86.

DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1007/s11854-009-0018-z
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Abstract

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.

Item Type: Journal Article
Copyright Holders: 2009 The Hebrew University Magnes Press, Jerusalem
ISSN: 0021-7670
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 22425
Depositing User: Philip Rippon
Date Deposited: 28 Jul 2010 11:39
Last Modified: 05 Dec 2012 15:43
URI: http://oro.open.ac.uk/id/eprint/22425
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