### 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 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Conformal Geometry and Dynamics*,

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

}

*Conformal Geometry and Dynamics*, vol. 15, no. 4, pp. 44-49. https://doi.org/10.1090/S1088-4173-2011-00224-4

**The schwarzian operator : Sequences, fixed points and N-cycles.** / Zemyan, Stephen Michael.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The schwarzian operator

T2 - Sequences, fixed points and N-cycles

AU - Zemyan, Stephen Michael

PY - 2011/4/25

Y1 - 2011/4/25

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85009826999&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009826999&partnerID=8YFLogxK

U2 - 10.1090/S1088-4173-2011-00224-4

DO - 10.1090/S1088-4173-2011-00224-4

M3 - Article

VL - 15

SP - 44

EP - 49

JO - Conformal Geometry and Dynamics

JF - Conformal Geometry and Dynamics

SN - 1088-4173

IS - 4

ER -