Skip to main content
SpringerLink
Log in

Rational Ahlfors Functions

  • Published:
Constructive Approximation Aims and scope

Abstract

We study a problem of Jeong and Taniguchi to find all rational maps which are Ahlfors functions. We prove that the rational Ahlfors functions of degree two are characterized by having positive residues at their poles. We then show that this characterization does not generalize to higher degrees, with the help of a numerical method for the computation of analytic capacity. We also provide examples of rational Ahlfors functions in all degrees.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Ahlfors, L.: Bounded analytic functions. Duke Math. J. 14, 1–11 (1947)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bell, S.R., Kaleem, F.: The structure of the semigroup of proper holomorphic mappings of a planar domain to the unit disc. Comput. Methods Funct. Theory 8, 225–242 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bell, S.R., Deger, E., Tegtmeyer, T.: A Riemann mapping theorem for two-connected domains in the plane. Comput. Methods Funct. Theory 9, 323–334 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bieberbach, L.: Über einen Riemannschen Satz aus der Lehre von der konformen Abbildung. Ber. Berliner Math. Ges. 24, 6–9 (1925)

    Google Scholar 

  5. Fisher, S.D.: On Schwarz’s lemma and inner functions. Trans. Am. Math. Soc 138, 229–240 (1969)

    MATH  Google Scholar 

  6. Garabedian, P.R.: Schwarz’s lemma and the Szegö kernel function. Trans. Am. Math. Soc. 67, 1–35 (1949)

    MATH  MathSciNet  Google Scholar 

  7. Garnett, J.: Analytic Capacity and Measure. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  8. Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable. American Mathematical Society, Providence (1969)

    MATH  Google Scholar 

  9. Grunsky, H.: Lectures on Theory of Functions in Multiply Connected Domains. Vandenhoeck & Ruprecht, Göttingen (1978)

    MATH  Google Scholar 

  10. Havinson, S.Ya.: Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains. Am. Math. Soc. Transl. 43, 215–266 (1964)

  11. Jeong, M., Taniguchi, M.: Bell representations of finitely connected planar domains. Proc. Am. Math. Soc. 131, 2325–2328 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jeong, M., Taniguchi, M.: The coefficient body of Bell representations of finitely connected planar domains. J. Math. Anal. Appl. 295, 620–632 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Khavinson, D.: On removal of periods of conjugate functions in multiply connected domains. Mich. Math. J. 31, 371–379 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  14. Newman, M.H.A.: Elements of the Topology of Plane Sets of Points. Cambridge University Press, New York (1961)

    Google Scholar 

  15. Younsi, M., Ransford, T.: Computation of analytic capacity and applications to the subadditivity problem. Comput. Methods Funct. Theory 13, 337–382 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank Thomas Ransford and Joe Adams for helpful discussions related to this work, as well as the referees for their useful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, The Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    Maxime Fortier Bourque

  2. Département de Mathématiques et de Statistique, Université Laval, Pavillon Alexandre-Vachon, 1045 av. de la Médecine, Québec, QC, G1V 0A6, Canada

    Malik Younsi

Authors
  1. Maxime Fortier Bourque
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Malik Younsi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Malik Younsi.

Additional information

Communicated by Doron S. Lubinsky.

Maxime Fortier Bourque was supported by NSERC. Malik Younsi was supported by the Vanier Canada Graduate Scholarships program.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fortier Bourque, M., Younsi, M. Rational Ahlfors Functions. Constr Approx 41, 157–183 (2015). https://doi.org/10.1007/s00365-014-9260-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-014-9260-4

Keywords

Mathematics Subject Classification

Search

Navigation