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UNIVALENT FUNCTIONS AND RADIAL GROWTH

Part of: Geometric function theory

Published online by Cambridge University Press:  16 January 2015

J. B. Twomey
J. B. Twomey*
Affiliation:
Department of Mathematics, University College Cork, Western Road Cork, Ireland email twomeyjb@ucc.ie
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Abstract

We address a question raised by Anderson, Hayman and Pommerenke relating to a classical result on univalent functions $f$ in the unit disc due to Spencer, and involving the size of the set of ${\it\theta}\in [-{\it\pi},{\it\pi}]$ for which we have $\log |f(r\text{e}^{\text{i}{\it\theta}})|\neq o(\log (1/(1-r)))$ as $r\rightarrow 1.$ An answer is given in terms of a certain generalized capacity, and also in terms of Hausdorff measure. Further results regarding the radial growth of univalent functions are also established, and some examples are constructed which relate to the sharpness of these results.

MSC classification

Primary: 30C35: General theory of conformal mappings
Type
Research Article
Information
Mathematika , Volume 61 , Issue 3 , September 2015 , pp. 531 - 546
Copyright
Copyright © University College London 2015 

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