### 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.

Original language | English (US) |
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Pages (from-to) | 44-49 |

Number of pages | 6 |

Journal | Conformal Geometry and Dynamics |

Volume | 15 |

Issue number | 4 |

DOIs | |

State | Published - Apr 25 2011 |

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

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## Cite this

Zemyan, S. M. (2011). The schwarzian operator: Sequences, fixed points and N-cycles.

*Conformal Geometry and Dynamics*,*15*(4), 44-49. https://doi.org/10.1090/S1088-4173-2011-00224-4