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Conformal Geometry and Dynamics

ISSN 1088-4173



The Schwarzian operator: sequences, fixed points and $N$-cycles

Author: Stephen M. Zemyan
Journal: Conform. Geom. Dyn. 15 (2011), 44-49
MSC (2010): Primary 34L30; Secondary 30D30, 34A25, 34A34
Published electronically: April 25, 2011
MathSciNet review: 2801171
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Abstract: Given a function $f(z)$ that is analytic in a domain $D$, we define the classical Schwarzian derivative $\{f,z\}$ of $f(z)$, and mention some of its most useful analytic properties. We explain how the process of iterating the Schwarzian operator produces a sequence of Schwarzian derivatives, and we illustrate this process with examples. Under a suitable restriction, these sequences become $N$-cycles of Schwarzian derivatives. Some properties of functions belonging to an $N$-cycle are listed. We conclude the article with a collection of related open problems.

References [Enhancements On Off] (What's this?)

  • Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197

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Additional Information

Stephen M. Zemyan
Affiliation: Department of Mathematics, Penn State Mont Alto, Mont Alto, Pennsylvania 17237-9799

Keywords: Schwarzian derivative
Received by editor(s): December 9, 2010
Published electronically: April 25, 2011
Article copyright: © Copyright 2011 American Mathematical Society