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

DOI:
https://doi.org/10.1090/S0002-9939-00-05623-9

Published electronically:
April 28, 2000

MathSciNet review:
1706981

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Abstract | References | Similar Articles | Additional Information

Let 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 . 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.

**1.**Bertilsson, Daniel.*Coefficient estimates for negative powers of the derivative of univalent functions*. Ark. Mat.**36**(1998), no. 2, 255-273. MR**99i:30030****2.**Harmelin, Reuven.*Aharanov invariants and univalent functions*. Israel J. Math**43**(1982) 3 pp 244-254. MR**85a:30020****3.**Lavie, Meira.*The Schwarzian derivative and disconjugacy of th order linear differential equations*. Canad. J. Math.**21**(1969).**4.**Kim, Seong-A and Minda, David.*Two point distortion theorems for univalent functions*. Pacific J. of Math.**163**1 pp 137-157. MR**94m:30042****5.**Nehari, Zeev.*The Schwarzian derivative and schlicht functions*. Bull. Amer. Math. Soc.**55**(1949) pp 545-551. MR**10:696e****6.**Pommerenke, Christian.*Univalent Functions*. Vandenhoeck and Ruprecht, Göttingen 1975. MR**58:22526****7.**Rosenblum, Marvin and Rovnyak, James.*Topics in Hardy Classes and Univalent Functions*. Birkhäuser Advanced Texts, 1994. MR**97a:30047****8.**Schober, Glenn.*Coefficient estimates for inverses of schlicht functions*. In Aspects of Contemporary Complex Analysis, ed. Brannan, D.A. and Clunie, J.G. 1980, pp. 503-513. MR**82j:30021****9.**Tamanoi, Hirotaka.*Higher Schwarzian operators and combinatorics of the Schwarzian derivative*. Math. Ann.**305**(1996) 1 pp 127-151. MR**97h:30001**

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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:
https://doi.org/10.1090/S0002-9939-00-05623-9

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