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Growth of functions in cercles de remplissage

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

P. C. Fenton  and
John Rossi
P. C. Fenton
Affiliation:
Department of Mathematics, University of Otago, Dunedin, New Zealand e-mail: pfenton@maths.otago.ac.nz
John Rossi
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg VA 24060, USA e-mail: rossi@calvin.math.vt.edu
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Abstract

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Suppose that f is meromorphic in the plane, and that there is a sequence Zn → ∞ and a sequence of positive numbers ∈n → 0, such that ∈n|zn|f#(zn)/log|zn| → ∞. It is shown that if f is analytic and non-zero in the closed discs Δn = {z: |z – zn| ≦∈n|zn|}, n = 1, 2, 3 …, then, given any positive integer K, there are arbitrarily large values of n and there is a point z in Δn such that │f (z)| 〉 |Z│k. Examples are given to show that the hypotheses cannot be relaxed.

MSC classification

Secondary: 30D20: Entire functions, general theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 1 , February 2002 , pp. 131 - 136
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Fenton, P. C. and Rossi, J., ‘Cercles de remplissage for entire function’, Bull. London Math. Soc. 31 (1999), 59–66.CrossRefGoogle Scholar
[2]Lehto, O., ‘The spherical derivative of a meromorphic function in the neighbourhood of an isolated singularity’, Comment. Math. Helv. 33 (1959), 196–205.CrossRefGoogle Scholar
[3]Rossi, J., ‘A sharp result concerning Cercles de remplissage’, Ann. Acad. Sc. Fenn. 20 (1995), 179–185.Google Scholar
[4]Titchmarsh, E. C., Theory of functions, 2nd edition (Oxford University Press, London, 1939).Google Scholar
[5]Yang, L., ‘Meromorphic functions and their derivatives’, J. London Math. Soc. 25 (1982), 288–296.Google Scholar