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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Growth and covering theorems associated with the Roper-Suffridge extension operator
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by Ian Graham PDF
Proc. Amer. Math. Soc. 127 (1999), 3215-3220 Request permission

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
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 3215-3220
  • MSC (1991): Primary 32H02; Secondary 30C25
  • DOI: https://doi.org/10.1090/S0002-9939-99-04963-1
  • MathSciNet review: 1618678