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Some results of Lindelöf type involving the segmental behaviour of holomorphic functions

Published online by Cambridge University Press:  24 October 2008

Francisca Bravo  and
Daniel Girela
Francisca Bravo
Affiliation:
I.F.P. Carlos Castilla del Pino, San Roque, Cádiz, Spain
Daniel Girela
Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
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Abstract

A classical theorem of Lindelöf asserts that if ƒ is a function analytic and bounded in the unit disc δ which has the asymptotic value L at a point ξ ε ∂ δ then it has the non-tangential limit L at ξ. This result does not remain true for functions f analytic in δ whose maximum modulus grows to infinity arbitrarily slowly. However, the second author has recently obtained some results of Lindelöf type valid for these functions. In this paper we obtain new results of this kind. We prove that if f is an analytic function of slow growth in δ and ξ ε ∂ δ, then certain restrictions on the growth of ƒ′ along a segment which ends at ξ do imply that ƒ has a non-tangential limit at ξ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 57 - 65
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

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