The Normality of Meromorphic Functions with Multiple Zeros and Poles Concerning Sharing Values

  • WANG, YOU-MING (Department of Applied Mathematics, College of Science, Hunan Agricultural University)
  • Received : 2013.10.22
  • Accepted : 2013.12.13
  • Published : 2015.09.23


In this paper we study the problem of normal families of meromorphic functions concerning shared values. Let F be a family of meromorphic functions in the plane domain $D{\subseteq}{\mathbb{C}}$ and n, k be two positive integers such that $n{\geq}k+1$, and let a, b be two finite complex constants such that $a{\neq}0$. Suppose that (1) $f+a(f^{(k)})^n$ and $g+a(g^{(k)})^n$ share b in D for every pair of functions f, $g{\in}F$; (2) All zeros of f have multiplicity at least k + 2 and all poles of f have multiplicity at least 2 for each $f{\in}F$ in D; (3) Zeros of $f^{(k)}(z)$ are not the b points of f(z) for each $f{\in}F$ in D. Then F is normal in D. And some examples are provided to show the result is sharp.


meromorphic functions;shared value;normal family


  1. W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Revista Matemtica Iberoamericana, 11(2)(1995), 355-375.
  2. H. H. Chen and Y. X. Gu, Y, An improvement of Marty criterion and its application, Sci China Ser A, 36(1993), 674-681.
  3. J. Cline and Y. K. Hayman, The spherical derivatives of integral and meromorphic functins, Comment Math Hev, 40(1996), 117-148.
  4. G. Datt and S. Kumar, Normality of meromorphic functions with multiple zeros and poles, arXiv:1305.6214V1.
  5. H. K. Hayman, Meromorphic functions, Clarendon, Oxford, 1964.
  6. X. C. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc, 32(2000), 325-331.
  7. D. W. Meng and P. C. Hu, Normality criteria of meromorphic functons sharing one value, J. Math. Anal. Appl., 381(2011), 724-731.
  8. J. Schiff, Normal families, Springer-Verlag, Berlin, 1993.
  9. W. Schwick, Sharing Values and Normality, Archiv der Mathematik, 59(1992),50-54.
  10. Y. Xu, F. Q. Wu and L. W. Liao, Picard values and normal families of meromorphic functions, Proc. Roy. Soc. Edinburgh Sect A, 139(2009), 1091-1099.
  11. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Science Press/Kluwer Academic, Beijing/New York, 2003.
  12. S. Zeng and I. Lahiri, A normality criterion for meromorphic functions, Kodai Math. J., 35(2012), 105-114.