Advertisement

Acta Mathematica Sinica, English Series

, Volume 35, Issue 12, pp 1972–1978 | Cite as

A Normal Criterion Concerning Omitted Holomorphic Function

  • Jin Hua YangEmail author
  • Qi YangEmail author
  • Xue Cheng PangEmail author
Article
  • 20 Downloads

Abstract

In this paper, we continue to discuss the normality concerning omitted holomorphic function and get the following result. Let \(\mathcal{F}\) be a family of meromorphic functions on a domain D, k ≥ 4 be a positive integer, and let a(z) and b(z) be two holomorphic functions on D, where a(z) ≢ 0 and f(z) ≢ ∞ whenever a(z) = 0. If for any \(f \in \mathcal{F}\), f′(z) − a(z)fk(z) ≠ b(z), then \(\mathcal{F}\) is normal on D.

Keywords

Normal family holomorphic functions omitted functions 

MR(2010) Subject Classification

30D35 30D45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We would like to thank Professor Mingliang Fang for raising the questions made to this paper.

References

  1. [1]
    Bergweiler, W., Eremenko, A.: On the singularities of the inrerse to a meromorphic function of finite order. Rev. Mate. Iberoamericana, 11, 355–373 (1995)Google Scholar
  2. [2]
    Bergweiler, W., Pang, X. C.: On the derivative of meromorphic functions with multiple zeros. J. Math. Anal. Appl., 278, 285–292 (2003)MathSciNetCrossRefGoogle Scholar
  3. [3]
    Chen, H. H., Fang, M. L.: The value distribution of f nf . Science in China, Series A, 38(7), 789–798 (1995)MathSciNetGoogle Scholar
  4. [4]
    Fang, M. L., Yuan, W. J.: On the normality for families of meromorphic functions. Indian J. Math., 43, 341–350 (2001)MathSciNetzbMATHGoogle Scholar
  5. [5]
    Gu, Y. X., Pang, X. C., Fang, M. L.: The Theory of Normal Family and Its Application, Science Press, Beijing, 2007Google Scholar
  6. [6]
    Hayman, W. K.: Picard values of meromorphic functions and their derivatives. Ann. of Math., 70(2), 9–42 (1959)MathSciNetzbMATHGoogle Scholar
  7. [7]
    Huang, X. J., Gu, Y. X.: Normal families of meromorphic functions. Result. Math., 49, 279–288 (2006)MathSciNetCrossRefGoogle Scholar
  8. [8]
    Li, X. J.: Proof of Hayman’s conjecture on normal families. Science in China, Series A, 28(6), 24–31 (1985)MathSciNetGoogle Scholar
  9. [9]
    Mues, E.: Über ein problem von Hayman. Math. Z., 163(3), 239–259 (1979)zbMATHGoogle Scholar
  10. [10]
    Pang, X. C.: Criteria for normality about differential polynomial. Chinese Science Bull., 33(22), 1690–1693 (1988)CrossRefGoogle Scholar
  11. [11]
    Pang, X. C.: On normal criterion of meromorphic functions. Science in China, Series A, 33(5), 521–527 (1990)MathSciNetGoogle Scholar
  12. [12]
    Pang, X. C., Nevo, S., Zalcman, L.: Quasinormal families of meromorphic functions II. Advances and Applications, 158, 177–189 (2005)MathSciNetGoogle Scholar
  13. [13]
    Pang, X. C., Zalcman, L.: Normal families of meromorphic functions with multiple zeros and poles. Israel J. Math., 136, 1–9 (2003)MathSciNetCrossRefGoogle Scholar
  14. [14]
    Pang, X. C., Yang, D. G., Zalcman, L.: Normal families of meromorphic functions whose derivatives omit a function. Comput. Meth. Funct. Th., 2(1), 257–265 (2002)MathSciNetGoogle Scholar
  15. [15]
    Pang, X. C., Zalcman, L.: Normality families and shared values. Bull. London Math. Soc., 32, 325–331 (2000)MathSciNetCrossRefGoogle Scholar
  16. [16]
    Zhang G. M., Pang, X. C., Zalcman, L.: Normal families and omitted functions II. Bull. London Math. Soc., 41, 63–71 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesXinjiang Normal UniversityUrumqiP. R. China
  2. 2.Department of MathematicsEast China Normal UniversityShanghaiP. R. China

Personalised recommendations