Arkiv för Matematik

  • Ark. Mat.
  • Volume 43, Number 2 (2005), 307-321.

Normality and fixed-points of meromorphic functions

Jianming Chang, Mingliang Fang, and Lawrence Zalcman

Full-text: Open access


LetF be families of meromorphic functions in a domain D, and let R be a rational function whose degree is at least 3. If, for any f∈F, the composite function R(f) has no fixed-point in D, thenF is normal in D. The number 3 is best possible. A new and much simplified proof of a result of Pang and Zalcman concerning normality and, shared values is also given.

Article information

Ark. Mat., Volume 43, Number 2 (2005), 307-321.

Received: 8 January 1994
Revised: 30 May 1994
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2005 © Institut Mittag-Leffler


Chang, Jianming; Fang, Mingliang; Zalcman, Lawrence. Normality and fixed-points of meromorphic functions. Ark. Mat. 43 (2005), no. 2, 307--321. doi:10.1007/BF02384782.

Export citation


  • Essén, M. and Wu, S., Fix-points and a normal family of analytic functions, Complex Variables Theory Appl. 37 (1998), 171–178.
  • Fang, M. L. and Xu, Y., Normal families of holomorphic functions and shared values, Israel J. Math. 129 (2002), 125–141.
  • Fang, M. L. and Yuan W. J., On Rosenbloom's fixed-point theorem and related results, J. Austral. Math. Soc. 68 (2000), 321–333.
  • Fang, M. L. and Zalcman, L., Normal families and shared values of meromorphic functions, Ann. Polon. Math. 80 (2003), 133–141.
  • Gross, F. and Osgood, C. F., On the fixed points of composite meromorphic functions, J. Math. Anal. Appl. 114 (1986), 490–496.
  • Hayman, W. K., Meromorphic Functions, Clarendon Press, Oxford, 1964.
  • Pang, X. C. and Zalcman, L., Normality and shared values, Ark. Mat. 38 (2000), 171–182.
  • Pang, X. C. and Zalcman, L., Normal families and shared values, Bull. London Math. Soc. 32 (2000), 325–331.
  • Rosenbloom, P. C., The fix-points of entire functions, Medd. Lunds Univ. Mat. Sem. 1952 (Tome Supplementaire) (1952), 186–192.
  • Schiff, J. L., Normal Families, Springer, New York, 1993.
  • Schwick, W., Sharing values and normality, Arch. Math. (Basel) 59 (1992), 50–54.
  • Yang, L., Normal families and fix-points of meromorphic functions, Indiana Univ. Math. J. 35 (1986), 179–191.
  • Yang, L., Some recent results and problems in the theory of value distribution, in Proc. of the Symposium on Value Distribution Theory in Several Complex Variables (Notre Dame, IN, 1990), Notre Dame Math. Lectures 12, pp. 157–171, Univ. of Notre Dame Press, Notre Dame, IN, 1992.
  • Yang, L., Value Distribution Theory. Translated and revised from the 1982 Chinese original, Springer, Berlin: Science Press, Beijing, 1993.