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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Iteration of a class of hyperbolic
meromorphic functions


Authors: P. J. Rippon and G. M. Stallard
Journal: Proc. Amer. Math. Soc. 127 (1999), 3251-3258
MSC (1991): Primary 30D05
Published electronically: April 27, 1999
MathSciNet review: 1610785
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Abstract | References | Similar Articles | Additional Information

Abstract: We look at the class $B_n$ which contains those transcendental meromorphic functions $f$ for which the finite singularities of $f^{-n}$ are in a bounded set and prove that, if $f$ belongs to $B_n$, then there are no components of the set of normality in which $f^{mn}(z)\to\infty$ as $m\to\infty$. We then consider the class $\widehat B$ which contains those functions $f$ in $B_1$ for which the forward orbits of the singularities of $f^{-1}$ stay away from the Julia set and show (a) that there is a bounded set containing the finite singularities of all the functions $f^{-n}$ and (b) that, for points in the Julia set of $f$, the derivatives $(f^n)'$ have exponential-type growth. This justifies the assertion that $\widehat B$ is a class of hyperbolic functions.


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Additional Information

P. J. Rippon
Affiliation: Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England
Email: p.j.rippon@open.ac.uk

G. M. Stallard
Affiliation: Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England
Email: g.m.stallard@open.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04942-4
PII: S 0002-9939(99)04942-4
Received by editor(s): September 30, 1997
Received by editor(s) in revised form: January 26, 1998
Published electronically: April 27, 1999
Dedicated: Dedicated to Professor Noel Baker on the occasion of his retirement
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society