Skip to main content
Log in

Schwarzian derivatives, the Poincaré metric and the kernel function

  • Published:
Commentarii Mathematici Helvetici

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

Access this article

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

Instant access to the full article PDF.

References

  1. L. V. Ahlfors, Quasiconformal reflections,Acta Math. 109 (1963) 291–301.

    Article  MATH  MathSciNet  Google Scholar 

  2. —,Conformal invariants: topics in geometric function theory. McGraw-Hill, New York 1973.

    MATH  Google Scholar 

  3. L. V. Ahlfors andA. Beurling, Conformal invariants and function-theoretic null-sets,Acta Math. 83 (1950) 101–129.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. F. Beardon and Ch.Pommerenke, The Poincaré metric of plane domains,J. London Math. Soc. (2), 18 (1978), 475–483.

    MATH  MathSciNet  Google Scholar 

  5. S. Bergman,The kernel function and conformal mapping. Amer. Math. Soc. Survey5, New York 1950.

  6. S. Bergman andM. Schiffer, Kernel functions and conformal mapping,Compositio Math. 8 (1951) 205–249.

    MATH  MathSciNet  Google Scholar 

  7. B. Epstein,Orthogonal families of analytic functions. Macmillan, New York 1965.

    MATH  Google Scholar 

  8. F. W. Gehring, Inequalities for condensers, hyperbolic capacity and extremal lengths,Mich. Math. J. 18 (1971) 1–20.

    Article  MATH  MathSciNet  Google Scholar 

  9. —, Univalent functions and the Schwarzian derivative,Comment. Math. Helvetici 52 (1977) 561–572.

    MATH  MathSciNet  Google Scholar 

  10. D. A. Hejhal, Universal covering maps for variable regions,Math. Zeitschr. 137 (1974) 7–20.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Osgood, Univalence criteria in multiply connected domains,Trans. Amer. Math. Soc. (to appear).

  12. M. Sakai, The sub-mean-value property of subharmonic functions and its application to estimates for the Gaussian curvature of the span metric,Hiroshima Math. J. (to appear).

  13. K. Zarankiewicz, Über ein numerisches Verfahren zur konformen Abbildung zweifach zusammenhängender Gebiete,Zeit. angew. Math. Mech. 14 (1934) 97–104.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dedicated to the memory of Professor Z. Nehari

This research was supported in part by the U.S. National Science Foundation, Grant MCS-77-02842.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beardon, A.F., Gehring, F.W. Schwarzian derivatives, the Poincaré metric and the kernel function. Commentarii Mathematici Helvetici 55, 50–64 (1980). https://doi.org/10.1007/BF02566674

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02566674

Keywords

Navigation