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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Distortion theorems for higher order Schwarzian derivatives of univalent functions


Author: Eric Schippers
Journal: Proc. Amer. Math. Soc. 128 (2000), 3241-3249
MSC (1991): Primary 30C55
Published electronically: April 28, 2000
MathSciNet review: 1706981
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Abstract:

Let $\tilde{\mathcal{S}}$ denote the class of functions which are univalent and holomorphic on the unit disc. We derive a simple differential equation for the Loewner flow of the Schwarzian derivative of a given $f \in \tilde{\mathcal{S}}$. This is used to prove bounds on higher order Schwarzian derivatives which are sharp for the Koebe function. As well we prove some two-point distortion theorems for the higher order Schwarzians in terms of the hyperbolic metric.


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Additional Information

Eric Schippers
Affiliation: Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, Canada M5S 3G3
Email: eric@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05623-9
PII: S 0002-9939(00)05623-9
Keywords: Schwarzian derivative, univalent functions, hyperbolic metric
Received by editor(s): December 14, 1998
Published electronically: April 28, 2000
Additional Notes: This paper is part of thesis work at the University of Toronto.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society