Skip to main content
SpringerLink
Log in

Families of meromorphic functions avoiding continuous functions

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

Leth 1,h 2 andh 3 be continuous functions from the unit disk D into the Riemann sphereC such thath i(z) ≠ hj(z) (i ≠ j) for eachz∈D. We prove that the setF of all functionsf meromorphic on D such thatf(z)≠h j (z) for allz ∈ D andj=1,2,3 is a normal family. The result and the method of the proof extend to quasimeromorphic functions in higher dimensions as well.

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

Access this article

Log in via an institution

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. L. V. Ahlfors,Complex Analysis, 3rd edn. McGraw-Hill, New York, 1979.

    MATH  Google Scholar 

  2. L. Bers and H. Royden,Holomorphic families of injections, Acta Math.157 (1986), 259–286.

    Article  MATH  MathSciNet  Google Scholar 

  3. K. Deimling,Nonlinear Functional Analysis, Springer, Berlin, 1985.

    MATH  Google Scholar 

  4. W. K. Hayman,Meromorphic Functions, Clarendon Press, Oxford, 1964.

    MATH  Google Scholar 

  5. W. K. Hayman and L. A. Rubel,Unavoidable systems of functions, Math. Proc. Cambridge Philos. Soc.117 (1995), 345–351.

    Article  MATH  MathSciNet  Google Scholar 

  6. O. Lehto and K. I. Virtanen,Quasiconformal Mappings in the Plane, Springer, Berlin, 1973.

    MATH  Google Scholar 

  7. R. Miniowitz,Normal Families of quasimeromorphic mappings, Proc. Amer. Math. Soc.84 (1982), 35–43.

    Article  MATH  MathSciNet  Google Scholar 

  8. P. Montel,Sur les familles de fonction analytiques qui admettent des valeurs exceptionnelles dans un domaine, Ann. Sci. École Norm. Sup. (3)29 (1912), 487–535.

    MathSciNet  Google Scholar 

  9. S. Rickman,Quasiregular Mappings, Springer, Berlin, 1993.

    MATH  Google Scholar 

  10. L. Zalcman,A heuristic principle in complex function theory, Amer. Math. Monthly82 (1975), 813–817.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. M. Bonk

    Present address: Department of Mathematics, University of Michigan, 48109, Ann Arbor, MI, USA

Authors and Affiliations

  1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098, Kiel, Germany

    D. Bargmann

  2. Institut für Analysis, Tu Braunschweig, Pockelsstr. 14, D-38106, Braunschweig, Germany

    M. Bonk

  3. Department of Mathematics, University of Illinois, 61801, Urbana, IL, USA

    A. Hinkkanen

  4. Department of Mathematics, University of Auckland, 92019, Auckland, New Zealand

    G. J. Martin

Authors
  1. D. Bargmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Bonk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Hinkkanen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. J. Martin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The second author was supported by a Heisenberg fellowship of the DFG. The fourth author was partially supported by the Marsden Fund, New Zealand. This research was completed while the authors were attending a conference at Mathematisches Forschungsinstitut Oberwolfach in Germany. The authors would like to express their sincere thanks to the Institute for providing a stimulating atmosphere and for its kind hospitality.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bargmann, D., Bonk, M., Hinkkanen, A. et al. Families of meromorphic functions avoiding continuous functions. J. Anal. Math. 79, 379–387 (1999). https://doi.org/10.1007/BF02788248

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02788248

Keywords

Search

Navigation