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

 

Growth and covering theorems associated
with the Roper-Suffridge extension operator


Author: Ian Graham
Journal: Proc. Amer. Math. Soc. 127 (1999), 3215-3220
MSC (1991): Primary 32H02; Secondary 30C25
Published electronically: April 27, 1999
MathSciNet review: 1618678
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Abstract | References | Similar Articles | Additional Information

Abstract: The Roper-Suffridge extension operator, originally introduced in the context of convex functions, provides a way of extending a (locally) univalent function $f\in \operatorname{Hol}(\mathbb{D},\mathbb{C})$ to a (locally) univalent map $F\in \operatorname{Hol}(B_{n},\mathbb{C}^{n})$. If $f$ belongs to a class of univalent functions which satisfy a growth theorem and a distortion theorem, we show that $F$ satisfies a growth theorem and consequently a covering theorem. We also obtain covering theorems of Bloch type: If $f$ is convex, then the image of $F$ (which, as shown by Roper and Suffridge, is convex) contains a ball of radius $\pi /4$. If $f\in S$, the image of $F$ contains a ball of radius $1/2$.


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Additional Information

Ian Graham
Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada M5S 1A1
Email: graham@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04963-1
PII: S 0002-9939(99)04963-1
Received by editor(s): January 20, 1998
Published electronically: April 27, 1999
Additional Notes: The author was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221.
Communicated by: Steven R. Bell
Article copyright: © Copyright 1999 American Mathematical Society