Skip to main content
Log in

Radially distributed values and normal families. II

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We consider the family of all functions holomorphic in the unit disk for which the zeros lie on one ray while the 1-points lie on two different rays. We prove that for certain configurations of the rays this family is normal outside the origin.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Bergweiler, A new proof of the Ahlfors five islands theorem, J. Anal. Math. 76 (1998), 337–347.

    Article  MathSciNet  Google Scholar 

  2. W. Bergweiler, Bloch’s principle, Comput. Methods Funct. Theory 6 (2006), 77–108.

    Article  MathSciNet  Google Scholar 

  3. W. Bergweiler and A. Eremenko, Radially distributed values and normal families. Int. Math. Res. Not. IMRN 2019 (2019), 7356–7378.

    Article  MathSciNet  Google Scholar 

  4. W. Bergweiler, A. Eremenko and A. Hinkkanen, Entire functions with two radially distributed values. Math. Proc. Cambridge Philos. Soc. 165 (2018), 93–108.

    Article  MathSciNet  Google Scholar 

  5. M. Biernacki, Sur la théorie des fonctions entières, Bulletin de l’Acadèmie Polonaise des Sciences et des Lettres, Classe des Sciences Mathèmatiques et Naturelles, Sèrie A (1929), 529–590.

    MATH  Google Scholar 

  6. A. Edrei, Meromorphic functions with three radially distributed values, Trans.Amer. Math. Soc. 78 (1955), 276–293.

    Article  MathSciNet  Google Scholar 

  7. A. Eremenko, Entire functions, PT-symmetry and Voros’s quantization scheme, preprint, arXiv: 1510.02504 [math-ph]

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

    Book  Google Scholar 

  9. L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, Berlin, 1990.

    MATH  Google Scholar 

  10. L. Hörmander, Notions of Convexity, Birkhhäuser, Boston, MA, 1994.

    MATH  Google Scholar 

  11. H. Milloux, Sur la distribution des valeurs des fonctions entieres d’ordre fini, à zéros reels, Bull. Sci. Math. (2) 51 (1927), 303–319.

    MATH  Google Scholar 

  12. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.

    Book  Google Scholar 

  13. T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.

    Book  Google Scholar 

  14. N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations. Springer, Cham, 2017.

    Book  Google Scholar 

  15. L. Zalcman, A heuristic principle in complex function theory. Amer. Math. Monthly 82 (1975), 813–817.

    Article  MathSciNet  Google Scholar 

  16. L. Zalcman, Normal families: new perspectives. Bull. Amer. Math. Soc. (N.S.) 35 (1998), 215–230.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Walter Bergweiler.

Additional information

Dedicated to Larry Zalcman

Supported by NSF grant DMS-1665115.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bergweiler, W., Eremenko, A. Radially distributed values and normal families. II. JAMA 141, 99–111 (2020). https://doi.org/10.1007/s11854-020-0116-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-020-0116-5

Navigation