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Rational Ahlfors Functions

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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.

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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 

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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.

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Correspondence to Malik Younsi.

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Communicated by Doron S. Lubinsky.

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

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

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

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