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Non-Tangential Limits of Slowly Growing Analytic Functions

Non-Tangential Limits of Slowly Growing Analytic Functions We show that if f is an analytic function in the unit disc $$\mathbb{D}$$ , $$M\left( {r,f} \right) = \mathcal{O}\left( {\left( {1 - r} \right)^{ - \eta } } \right) as r \to 1, for every \eta > 0,$$ , and $$\mathop {\sup }\limits_{0 \leqslant r < 1} (1 - r)^s \left| {f'\left( {r\zeta } \right) < \infty } \right|, where \left| \zeta \right| = 1, s < 1,$$ then f has a finite non-tangential limit at ζ. We also show that in this result it is not sufficient to assume that $$M\left( {r,f} \right) = \mathcal{O}\left( {\left( {1 - r} \right)^{ - \eta } } \right) as r \to 1, for some fixed \eta > 0.$$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Non-Tangential Limits of Slowly Growing Analytic Functions

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Publisher
Springer Journals
Copyright
Copyright © 2008 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321672
Publisher site
See Article on Publisher Site

Abstract

We show that if f is an analytic function in the unit disc $$\mathbb{D}$$ , $$M\left( {r,f} \right) = \mathcal{O}\left( {\left( {1 - r} \right)^{ - \eta } } \right) as r \to 1, for every \eta > 0,$$ , and $$\mathop {\sup }\limits_{0 \leqslant r < 1} (1 - r)^s \left| {f'\left( {r\zeta } \right) < \infty } \right|, where \left| \zeta \right| = 1, s < 1,$$ then f has a finite non-tangential limit at ζ. We also show that in this result it is not sufficient to assume that $$M\left( {r,f} \right) = \mathcal{O}\left( {\left( {1 - r} \right)^{ - \eta } } \right) as r \to 1, for some fixed \eta > 0.$$

Journal

Computational Methods and Function TheorySpringer Journals

Published: May 22, 2007

References