Skip to main content
SpringerLink
Log in

Quadrilaterals and extremal quasiconformal extensions

  • Published:
Commentarii Mathematici Helvetici

Abstract

We show that the smallest maximal dilatation for a quasiconformal extension of a quasisymmetric function of the unit circle may be larger than indicated by the change in the module of the quadrilaterals with vertices on the circle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agard, S. andKelingos, J. A.,On parametric representation for quasisymmetric functions, Comment. Math. Helv.44 (1969), 446–456.

    Article  MathSciNet  Google Scholar 

  2. Beurling, A. andAhlfors, L. V.,The boundary correspondence under quasiconformal mappings, Acta Math.96 (1956), 125–142.

    Article  MathSciNet  Google Scholar 

  3. Grötzsch, H.,Über möglichst konforme Abbildungen von schlichten Bereichen, Ber. Verh. Sächs. Akad. Wiss. Leipzig, math.-naturw. Kl.84 (1932), 114–120.

    Google Scholar 

  4. Jenkins, J. A.,On quasiconformal mappings with given boundary values, Indiana Univ. Math. J.37 (1988), 929–934.

    Article  MathSciNet  Google Scholar 

  5. Kelingos, J. A.,Boundary correspondence under quasiconformal mappings, Mich. Math. J.13 (1966), 235–249.

    Article  MathSciNet  Google Scholar 

  6. Lehto, O.,Univalent Functions and Teichmüller Spaces, Springer, New York, 1987.

    MATH  Google Scholar 

  7. Lehto, O. andVirtanen, K. I.,Quasiconformal Mappings in the Plane, Springer, Berlin, 1973.

    MATH  Google Scholar 

  8. Pólya, G. andSzegö, G.,Problems and Theorems in Analysis I, Springer, Berlin, 1972.

    MATH  Google Scholar 

  9. Reich, E.,Quasiconformal mappings of the disk with given boundary values, pp. 101–137 in Lecture Notes in Math. 505, Springer, Berlin, 1976.

    Google Scholar 

  10. Reich, E.,Uniqueness of Hahn—Banach extensions from certain spaces of analytic functions, Math. Z.167 (1979), 81–89.

    Article  MathSciNet  Google Scholar 

  11. Reich, E. andStrebel, K.,On the extremality of certain Teichmüller mappings, Comment. Math. Helv.45 (1970), 353–362.

    MathSciNet  Google Scholar 

  12. Strebel, K.,Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises II, Comment. Math. Helv.39 (1964), 77–89.

    MathSciNet  Google Scholar 

  13. Teichmüller, O.,Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss., math.-naturw. Kl.22 (1939), 3–197, also pp. 337–531 in Gesammelte Abhandlungen, Collected Works, Springer, Berlin, 1982.

    Google Scholar 

  14. Yang, S.,Extremal quasiconformal extensions, preprint, 1993.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University College, WC1E 6BT, London, U.K.

    J. M. Anderson & A. Hinkkanen

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, 61801, Urbana, Illinois, USA

    J. M. Anderson & A. Hinkkanen

Authors
  1. J. M. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Hinkkanen
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research of the second author has been partially supported by the Alfred P. Sloan Foundation and by the U.S. National Science Foundation grant DMS 94-00999.

This research was completed while the first author was visiting the University of Illinois at Urbana-Champaign. He wishes to thank the Department of Mathematics for its kind hospitality.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anderson, J.M., Hinkkanen, A. Quadrilaterals and extremal quasiconformal extensions. Commentarii Mathematici Helvetici 70, 455–474 (1995). https://doi.org/10.1007/BF02566018

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02566018

Keywords

Search

Navigation