Functions of small growth with no unbounded Fatou components |
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Authors: | P J Rippon G M Stallard |
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Institution: | 1.Department of Mathematics and Statistics,The Open University,Milton Keynes,UK |
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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. |
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Keywords: | |
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