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Hyperbolic linear invariance and hyperbolic k-convexity

Part of: Geometric function theory Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

Wancang Ma  and
David Minda
Wancang Ma
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.
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Abstract

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Pommerenke initiated the study of linearly invariant families of locally schlicht holomorphic functions defined on the unit disk The concept of linear invariance has proved fruitful in geometric function theory. One aspect of Pommerenke's work is the extension of certain results from classical univalent function theory to linearly invariant functions. We propose a definition of a related concept that we call hyperbolic linear invariance for locally schlicht holomorphic functions that map the unit disk into itself. We obtain results for hyperbolic linearly invariant functions which generalize parts of the theory of bounded univalent functions. There are many similarities between linearly invariant functions and hyperbolic linearly invariant functions, but some new phenomena also arise in the study of hyperbolic linearly invariant functions.

MSC classification

Secondary: 30C99: None of the above, but in this section 30C25: Covering theorems in conformal mapping theory 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30D45: Bloch functions, normal functions, normal families
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 1 , February 1995 , pp. 73 - 93
Copyright
Copyright © Australian Mathematical Society 1995

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