Journal d’Analyse Mathematique

, Volume 76, Issue 1, pp 337–347 | Cite as

A new proof of the Ahlfors five islands theorem

  • Walter Bergweiler


We deduce the Ahlfors five islands theorem from a corresponding result of Nevanlinna concerning perfectly branched values, a rescaling lemma for non-normal families and an existence theorem for quasiconformal mappings. We also give a proof of Nevanlinna’s result based on the rescaling lemma and a version of Schwarz’s lemma.


Meromorphic Function Existence Theorem Periodic Point Quasiconformal Mapping Quasiregular Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. V. Ahlfors,Sur une généralisation du théorème de Picard, C. R. Acad. Sci. Paris194 (1932), 245–247, andCollected Papers, Vol. I, BirkhÄuser, Boston, Basel, Stuttgart, 1982, pp. 145–147.MATHGoogle Scholar
  2. [2]
    L. V. Ahlfors,Sur les fonctions inverses des fonctions méromorphes, C. R. Acad. Sci. Paris194 (1932), 1145–1147, andCollected Papers, Vol. I, pp. 149–151.MATHGoogle Scholar
  3. [3]
    L. V. Ahlfors,Quelques propriétés des surfaces de Riemann correspondant aux fonctions méromorphes, Bull. Soc. Math. France60 (1932), 197–207, andCollected Papers, Vol. I, pp. 152–162.MATHMathSciNetGoogle Scholar
  4. [4]
    L. V. Ahlfors,über die Kreise die von einer Riemannschen FlÄche schlicht überdeckt werden, Comment. Math. Helv.5 (1933), 28–38, andCollected Papers, Vol. I, pp. 163–173.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    L. V. Ahlfors,Zur Theorie der überlagerungsflÄchen, Acta Math.65 (1935), 157–194, andCollected Papers, Vol. I, pp. 214–251.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    L. V. Ahlfors,An extension of Schwarz’s lemma, Trans. Amer. Math. Soc.43 (1938), 359–364, andCollected Papers, Vol. I, pp. 350–355.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    L. V. Ahlfors,Lectures on Quasiconformal Mappings, Van Nostrand, Princeton, 1966.MATHGoogle Scholar
  8. [8]
    L. V. Ahlfors,Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966.MATHGoogle Scholar
  9. [9]
    I. N. Baker,Repulsive fixpoints of entire functions, Math. Z.104 (1968), 252–256.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    I. N. Baker, J. Kotos and Y. Lü,Iterates of meromorphic functions, I, Ergodic Theory Dynam. Systems11 (1991), 241–248.MATHGoogle Scholar
  11. [11]
    D. Bargmann,Simple proofs of some fundamental properties of the Julia set, Ergodic Theory Dynam. Systems, to appear.Google Scholar
  12. [12]
    A. F. Beardon,Iteration of Rational Functions, Springer, New York, Berlin, Heidelberg, 1991.MATHGoogle Scholar
  13. [13]
    W. Bergweiler,Periodic points of entire functions: proof of a conjecture of Baker, Complex Variables Theory Appl.17 (1991), 57–72.MATHMathSciNetGoogle Scholar
  14. [14]
    A. Bloch,La conception actuelle de la théorie des fonctions entières et méromorphes, Enseign. Math.25 (1926), 83–103.Google Scholar
  15. [15]
    A. Bloch,Les fonctions holomorphes et méromorphes dans le cercle-unité, Gauthiers-Villars, Paris, 1926.MATHGoogle Scholar
  16. [16]
    A. Bolsch,Repulsive periodic points of meromorphic functions, Complex Variables Theory Appl.31 (1996), 75–79.MathSciNetGoogle Scholar
  17. [17]
    M. Bonk and A. Eremenko,Schlicht regions for entire and meromorphic functions, preprint, 1998.Google Scholar
  18. [18]
    C. Carathéodory,Sur quelques généralisations du théorème de M. Picard, C. R. Acad. Sci. Paris141 (1905), 1213–1215.Google Scholar
  19. [19]
    L. Carleson and T. W. Gamelin,Complex Dynamics, Springer, New York, Berlin, Heidelberg, 1993.MATHGoogle Scholar
  20. [20]
    P. Dominguez,Connectedness properties of Julia sets of transcendental entire functions, Complex Variables Theory Appl.32 (1997), 199–215.MATHMathSciNetGoogle Scholar
  21. [21]
    A. Eremenko,Bloch radius, normal families and quasiregular mappings, Proc. Amer. Math. Soc, to appear.Google Scholar
  22. [22]
    W. K. Hayman,Meromorphic Functions, Clarendon Press, Oxford, 1964.MATHGoogle Scholar
  23. [23]
    O. Lehto,Quasiconformal homeomorphisms and Beltrami equations, inDiscrete Groups and Automorphic Forms (W. J. Harvey, ed.), Academic Press, London, New York, San Francisco, 1977, pp. 121–142.Google Scholar
  24. [24]
    O. Lehto and K. I. Virtanen,Quasiconformal Mappings in the Plane, Springer, Berlin, Heidelberg, New York, 1973.MATHGoogle Scholar
  25. [25]
    A. J. Lohwater and Ch. Pommerenke,On normal meromorphic functions, Ann. Acad. Sci. Fenn., Ser. A., no. 550(1973).Google Scholar
  26. [26]
    Z. Nehari,A generalization of Schwarz lemma, Duke Math. J.14 (1947), 1035–1049.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    R. Nevanlinna,Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Gauthiers-Villars, Paris, 1929.MATHGoogle Scholar
  28. [28]
    R. Nevanlinna,Eindeutige analytische Funktionen, Springer, Berlin, Göttingen, Heidelberg, 1953.MATHGoogle Scholar
  29. [29]
    R. M. Robinson,A generalization of Picard’s and related theorems, Duke Math. J.5 (1939), 118–132.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    J. Schiff,Normal Families, Springer, New York, Berlin, Heidelberg, 1993.MATHGoogle Scholar
  31. [31]
    W. Schwick,Repelling periodic points in the Julia set, Bull. London Math. Soc.29 (1997), 314–316.MATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    G. M. Stallard,The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math. Soc. (2)49 (1994), 281–295.MATHMathSciNetGoogle Scholar
  33. [33]
    N. Steinmetz,Rational Iteration, Walter de Gruyter, Berlin, 1993.MATHGoogle Scholar
  34. [34]
    M. Tsuji,Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.MATHGoogle Scholar
  35. [35]
    G. Valiron,Families normales et quasi-normales de fonctions méromorphes, Gauthiers-Villars, Paris, 1929.Google Scholar
  36. [36]
    L. Zalcman,A heuristic principle in complex function theory, Amer. Math. Monthly82 (1975), 813–817.MATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    L. Zalcman,Normal families: new perspectives, Bull. Amer. Math. Soc.35 (1998), 215–230.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 1998

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrechts-UniversitÄt zu KielKielGermany

Personalised recommendations