Advertisement

On the Angular Derivative of Comb Domains

  • Nikolaos KaramanlisEmail author
Article

Abstract

The angular derivative problem is to provide geometric conditions on the boundary of a simply connected domain \(\Omega \) near \(\zeta \in \partial \Omega \) which are equivalent to the existence of a non-zero angular derivative at \(\zeta \) for the conformal map of \(\Omega \) onto the half plane. Rodin and Warschawski (Math Z 153:1–17, 1977), proposed a conjecture regarding the existence of an angular derivative at \(+\,\infty \) for a certain class of comb domains. The purpose of this paper is to show how a theorem of Burdzy (Math Z 192:89–107, 1986) can be used to give an affirmative answer to the necessity part of the Rodin–Warschawski conjecture on comb domains.

Keywords

Conformal map Angular derivative Lipschitz graphs Extremal distance 

Mathematics Subject Classification

30C35 31A15 

Notes

Acknowledgements

I would like to thank Prof. Don Marshall and Prof. Dimitrios Betsakos for all their help with this work. I would also like to thank the referees for their helpful suggestions.

References

  1. 1.
    Burdzy, K.: Brownian excursions and minimal thinness III. Math. Z. 192, 89–107 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carroll, T.F.: A classical proof of Burdzy’s theorem on the angular derivative. J. Lond. Math. Soc. 2(38), 423–441 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gardiner, S.J.: A short proof of Burdzy’s theorem on the angular derivative. Bull. Lond. Math. Soc. 23(6), 575–579 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Garnett, J.B., Marshall, D.E.: Harmonic Measure. New Mathematical Monographs, vol. 2. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  5. 5.
    Jenkins, J.A.: On comb domains. Proc. Am. Math. Soc. 124(1), 187–191 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jenkins, J.A.: The Method of the Extremal Metric. Handbook of Complex Analysis: Geometric Function Theory, Vol. 1, pp. 393–456. North-Holland, Amsterdam (2002)Google Scholar
  7. 7.
    Jenkins, J.A., Oikawa, K.: Conformality and semi-conformality at the boundary. J. für die reine und angewandte Math 291, 92–117 (1977)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Rodin, B., Warschawski, S.: Extremal length and the boundary behavior of conformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. 2, 467–500 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rodin, B., Warschawski, S.: Extremal length and univalent functions I. The angular derivative. Math. Z. 153, 1–17 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sastry, S.: Existence of an angular derivative for a class of strip domains. Proc. Am. Math. Soc. 123, 1075–1082 (1995)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece

Personalised recommendations