Rational Ahlfors Functions

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• 作者：Maxime Fortier?Bourque ; Malik Younsi
• 关键词：Analytic capacity ; Ahlfors functions ; Rational maps ; Conformal representation ; Primary 30C85 ; 30C20 ; Secondary 30F10 ; 65E05
• 刊名：Constructive Approximation
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：41
• 期：1
• 页码：157-183
• 全文大小：480 KB
• 参考文献：1. Ahlfors, L.: Bounded analytic functions. Duke Math. J. 14, 1-1 (1947) CrossRef
  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, 225-42 (2008) CrossRef
  3. Bell, S.R., Deger, E., Tegtmeyer, T.: A Riemann mapping theorem for two-connected domains in the plane. Comput. Methods Funct. Theory 9, 323-34 (2009) CrossRef
  4. Bieberbach, L.: über einen Riemannschen Satz aus der Lehre von der konformen Abbildung. Ber. Berliner Math. Ges. 24, 6- (1925)
  5. Fisher, S.D.: On Schwarz’s lemma and inner functions. Trans. Am. Math. Soc 138, 229-40 (1969)
  6. Garabedian, P.R.: Schwarz’s lemma and the Szeg? kernel function. Trans. Am. Math. Soc. 67, 1-5 (1949)
  7. Garnett, J.: Analytic Capacity and Measure. Springer, Berlin (1972) CrossRef
  8. Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable. American Mathematical Society, Providence (1969)
  9. Grunsky, H.: Lectures on Theory of Functions in Multiply Connected Domains. Vandenhoeck & Ruprecht, G?ttingen (1978)
  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, 215-66 (1964)
  11. Jeong, M., Taniguchi, M.: Bell representations of finitely connected planar domains. Proc. Am. Math. Soc. 131, 2325-328 (2003) CrossRef
  12. Jeong, M., Taniguchi, M.: The coefficient body of Bell representations of finitely connected planar domains. J. Math. Anal. Appl. 295, 620-32 (2004) CrossRef
  13. Khavinson, D.: On removal of periods of conjugate functions in multiply connected domains. Mich. Math. J. 31, 371-79 (1984) CrossRef
  14. Newman, M.H.A.: Elements of the Topology of Plane Sets of Points. Cambridge University Press, New York (1961)
  15. Younsi, M., Ransford, T.: Computation of analytic capacity and applications to the subadditivity problem. Comput. Methods Funct. Theory 13, 337-82 (2013) CrossRef
  
• 作者单位：Maxime Fortier?Bourque (1)
  Malik Younsi (2)
  
  1. Department of Mathematics, The Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA
  2. Département de Mathématiques et de Statistique, Université Laval, Pavillon Alexandre-Vachon, 1045 av. de la Médecine, Québec, QC, G1V 0A6, Canada
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Analysis
  
• 出版者：Springer New York
• ISSN：1432-0940

文摘

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.