Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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.

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

  • 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 $n$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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30C55

Retrieve articles in all journals with MSC (1991): 30C55

Additional Information

Eric Schippers
Affiliation: Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, Canada M5S 3G3

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

American Mathematical Society