Journal d’Analyse Mathématique

, Volume 77, Issue 1, pp 69–104

Schlicht regions for entire and meromorphic functions

  • M. Bonk
  • A. Eremenko


Let f:CC be a meromorpMc function. We study the size of the maximal disc inC, with respect to the spherical metric, in which a single-valued branch of f-1 exists. This problem is related to normality and type criteria. Best possible lower estimates of the size of such discs are obtained for entire functions and a class of meromorphic functions containing all elliptic functions. An estimate for the class of rational functions is also given which is best possible for rational functions of degree 7. For algebraic functions of given genus we obtain an estimate which is precise for genera 2 and 5 and asymptotically best possible when the genus tends to infinity.


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  1. [1]
    L. Ahlfors,Sur les fonctions inverses des fonctions méromorpkes, C. R. Acad. Sci. Paris194 (1932), 1145–1147.MATHGoogle Scholar
  2. [2]
    P. P. Belinskii,General Properties of Quasiconformal Mappings, Nauka, Siberian division, 1974 (in Russian).Google Scholar
  3. [3]
    W. Beigweiler and A. Eremenko,On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana11 (1995), 355–373.MathSciNetGoogle Scholar
  4. [4]
    A. Bloch,Les théorèmes de M. Valiron sur les fonctions entières at la théorie de l’uniformisation, Ann. Fac. Sci. Univ. Toulouse (3)17 (1926), 1–22.MathSciNetGoogle Scholar
  5. [5]
    R. Brody,Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc.235 (1978), 213–219.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    W. Chauvenet,A Treatise on Plane and Spherical Trigonometry, J. B. Lippincott Co, Philadelphia, 1850.Google Scholar
  7. [7]
    H. Chen,Yosida functions and Picard values of integral functions and their derivatives, Bull. Austral. Math. Soc.54 (1996), 373–381.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    J. Clunie and W. Hayman,The spherical derivative of integral and meromorphic functions, Comment. Math. Heiv.40 (1966), 117–148.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. de Guzman,Differentiation of integrals in Rn, Lecture Notes in Math.421, Springer, New York, 1975.Google Scholar
  10. [10]
    W. Hayman and P. Kennedy,Subharmonic Functions, Vol. 2, Academic Press, New York, 1989.MATHGoogle Scholar
  11. [11]
    E. Hille,Ordinary Differential Equations in the Complex Domain, Wiley, New York, 1976.MATHGoogle Scholar
  12. [12]
    S. Lang,Introduction to Complex Hyperbolic Spaces, Springer, Berlin, 1987.MATHGoogle Scholar
  13. [13]
    O. Lehto and K. Virtanen,Quasikonforme Abbildungen, Springer, Berlin, 1965.MATHGoogle Scholar
  14. [14]
    A. Lohwater and Ch. Pommerenke,On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math.550 (1973), 1–12.Google Scholar
  15. [15]
    D. Minda,Euclidean, hyperbolic and spherical Block constants, Bull. Amer. Math. Soc.6 (1982), 441–444.MATHMathSciNetGoogle Scholar
  16. [16]
    D. Minda,Bloch constants for memmorphic functions, Math. Z.181 (1982), 83–92.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    D. Minda,Yosida functions, inLectures on Complex Analysis (Chuang Chi-Tai, ed.), Proc. Symp. Complex Anal., World Scientific, London, 1988, pp. 197–213.Google Scholar
  18. [18]
    R. Nevaniirma,Analytic Functions, Springer, Berlin, 1970.Google Scholar
  19. [19]
    E. Peschl,über unverzweigte konforme Abbildungen, österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II185 (1976), 55–78.MathSciNetMATHGoogle Scholar
  20. [20]
    Ch. Pommerenke,Estimates for normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math.476 (1970), 1–12.Google Scholar
  21. [21]
    G. Valiron,Recherches sur les théorème de M. Picard, Ann. Sci. école Norm. Sup.38 (1921), 389–430.MathSciNetGoogle Scholar
  22. [22]
    L. I. VolkovysM,Converging sequences of Riemann surfaces, Mat. Sb.23, 3 (1948), 361–382 (in Russian); Engl. transi.: Amer. Math. Soc. Transi. Ser. 2, Vol. 32.Google Scholar
  23. [23]
    L. I. Volkovyskii,Researches on the type problem of a simply connected Riemann surface, Proc. Steklov Inst. Math., Acad. Sci. USSR34 1950 (in Russian).Google Scholar
  24. [24]
    H. Wittich,Neuere Untersuchungen über eindeutige analytische Funktionen, Springer, Berlin, 1955.MATHGoogle Scholar
  25. [25]
    K. Yosida,On a class of meromorphic functions, Proc. Phys.-Math. Soc. Japan16 (1934), 227–235.MATHGoogle Scholar
  26. [26]
    L. Zalcman,A heuristic principle in complex function theory, Amer. Math. Monthly82 (1975), 813–817.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    L. Zalcman,Normal families: New perspectives, Bull. Amer. Math. Soc.35 (1998), 215–230.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 1999

Authors and Affiliations

  • M. Bonk
    • 1
  • A. Eremenko
    • 2
  1. 1.Fachbereich Mathematik MA 8-2Technische UniversitÄt BerlinBerlinGermany
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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