Skip to main content
SpringerLink
Log in

Faber polynomials with applications to univalent functions with quasiconformal extensions

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

We obtain some convergence properties concerning Faber polynomials and apply them to studying univalent functions with quasiconformal extensions. In particular, by introducing an operator on the usual l 2 space, we obtain some new characterizations of quasiconformal extendablity and asymptotic conformality for univalent functions.

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. Pommerenke Ch. Univalent Functions. Göttingen: Vandenhoeck and Ruprecht, 1975

    MATH  Google Scholar 

  2. Kühnau R. Verzerrungssätze und Koeffizientenbedingungen vom Grunskyschen typ für quasi-konforme abbildungen. Math Nachr, 48: 77–105 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cavus A. Approximation by generalized faber series in Bergman spaces on finite regions with a quasiconformal boundary. J Approx Theory, 87: 25–35 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Pommerenke Ch. Boundary Behaviour of Conformal Maps. Berlin: Springer-Verlag, 1992

    MATH  Google Scholar 

  5. Gardiner F P, Sullivan D. Symmetric structures on a closed curve. Amer J Math, 114: 683–736 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Earle C J, Gardiner F P, Lakic N. Asymptotic Teichmüller space, Part I: The complex structure. Contemp Math, 256: 17–38 (2000)

    MathSciNet  Google Scholar 

  7. Earle C J, Gardiner F P, Lakic N. Asymptotic Teichmüller space, Part II: The metric structure. Contemp Math, 355: 187–219 (2004)

    MathSciNet  Google Scholar 

  8. Earle C J, Markovic V, Saric D. Barycentric extension and the Bers embedding for asymptotic Teichmüller space. Contemp Math, 311: 87–105 (2002)

    MathSciNet  Google Scholar 

  9. Gardiner F P, Lakic N. Quasiconformal Teichmüller Theory. Math Surveys Monogr, 76. Providence, RI: Amer Math Soc, 2000

    Google Scholar 

  10. Kraus W. Über den Zusammenhang einiger Charrakteristiken eines einfach zusa- mmenhängenden Bereiches mit der Kreisabbildung. Mitt Math Sem Giessen, 21: 1–28 (1932)

    Google Scholar 

  11. Nehari Z. The Schwarzian derivative and schlicht functions. Bull Amer Math Soc, 55: 545–551 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kühnau R. Wertannahmeprobleme bei quasikonformen Abbildungen mit ortsab-hä-ngiger Dilatationsbeschränkung. Math Nachr, 40: 1–11 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lehto O. Schlicht functions with a quasiconformal extension. Ann Acad Sci Fenn A I Math, 500: 1–10 (1971)

    Google Scholar 

  14. Bazilevic I E. On a criterion of univalence of regular functions and the disposition of their coefficients. Math USSR Sb, 3: 123–137 (1967)

    Article  MathSciNet  Google Scholar 

  15. Zuravlev I V. Some sufficient conditions for the quasiconformal extension of analytic functions. Soviet Math Dokl, 19: 1549–1552 (1978)

    Google Scholar 

  16. Harmelin R. Bergman kernel function and univalence criteria. J Anal Math, 41: 249–258 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bergman S, Schiffer M M. Kernal functions and conformal mappings. Compos Math, 8: 205–249 (1951)

    MATH  MathSciNet  Google Scholar 

  18. Takhtajan L, Teo L P. Weil-Petersson metric on the universal Teichmüller space. Mem Amer Math Soc, 861: 1–183 (2006)

    MathSciNet  Google Scholar 

  19. Shen Y. On Grunsky operator. Sci China Ser A, 50: 1805–1817 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Becker J, Pommenerke Ch. Über die quasikonforme fortsetzung schlichter funktionen. Math Z, 161: 69–80 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  21. Schur I. Ein Satz Über quadratische Formen mit komplexen Koeffizienten. Amer J Math, 67: 472–480 (1945)

    Article  MATH  MathSciNet  Google Scholar 

  22. Schober G. Univalent functions-selected topics. Lecture Notes in Math, 478. Berlin: Springer-Verlag, 1975

    Google Scholar 

  23. Cui G. Integrably asymptotic affine homeomorphisms of the circle and Teichmüller spaces. Sci China Ser A, 43: 267–279 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Soochow University, Suzhou, 215006, China

    YuLiang Shen

Authors
  1. YuLiang Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YuLiang Shen.

Additional information

This work was supported by the Program for New Century Excellent Talents in University (Grant No. 06-0504) and National Natural Science Foundation of China (Grant No. 10771153)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shen, Y. Faber polynomials with applications to univalent functions with quasiconformal extensions. Sci. China Ser. A-Math. 52, 2121–2131 (2009). https://doi.org/10.1007/s11425-009-0062-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-009-0062-2

Keywords

MSC(2000)

Search

Navigation