Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Normality and exceptional values of derivatives

Author: Walter Bergweiler
Journal: Proc. Amer. Math. Soc. 129 (2001), 121-129
MSC (1991): Primary 30D45, 30D30
Published electronically: June 21, 2000
MathSciNet review: 1695100
Full-text PDF

Abstract | References | Similar Articles | Additional Information


We show that a family $\mathcal{F}$ of functions meromorphic in some domain is normal, if for all $f\in\mathcal{F}$ the derivative $f'$ omits the value $1$ and if the values that $f'$ can take at the zeros of $f$ satisfy certain restrictions. As an application we obtain a new proof of a theorem of Langley which classifies the functions $f$ meromorphic in the plane such that $f$ and $f''$ have no zeros.

References [Enhancements On Off] (What's this?)

  • 1. W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., New Ser. 29 (1993), 151-188. MR 94c:30033
  • 2. -, On the zeros of certain homogeneous differential polynomials, Arch. Math. 64 (1995), 199-202. MR 96a:30035
  • 3. W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoam. 11 (1995), 355-373. MR 96h:30055
  • 4. F. Brüggemann, Proof of a conjecture of Frank and Langley concerning zeros of meromorphic functions and linear differential polynomials, Analysis 12 (1992), 5-30. MR 93e:30067
  • 5. A. Eremenko and M. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier 42 (1992), 989-1020. MR 93k:30034
  • 6. G. Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z. 149 (1976), 29-36. MR 54:10601
  • 7. G. Frank and S. Hellerstein, On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. III. Ser. 53 (1986), 407-428. MR 88k:30032
  • 8. Yongxing Gu, On normal families of holomorphic functions (Chinese), Acta Math. Sinica 23 (1980), 157-161.
  • 9. W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. Math., II. Ser. 70 (1959), 9-42. MR 22:1675
  • 10. -, Meromorphic Functions, Clarendon Press, Oxford, 1964. MR 29:1337
  • 11. J. K. Langley, Proof of a conjecture of Hayman concerning $f$ and $f''$, J. London Math. Soc. II. Ser. 48 (1993), 500-514. MR 94k:30075
  • 12. -, On second order linear differential polynomials, Result. Math. 26 (1994), 51-82. MR 95k:30059
  • 13. J. K. Langley and J. H. Zheng, On the fixpoints, multipliers and value distribution of certain classes of meromorphic functions, Ann. Acad. Sci. Fenn., Math. 23 (1998), 133-150. MR 99b:30044
  • 14. Xuecheng Pang, Bloch's principle and normal criterion, Sci. China, Ser. A 32 (1989), 782-791. MR 91i:30031
  • 15. -, On normal criterion of meromorphic functions, Sci. China, Ser. A 33 (1990), 521-527. MR 92b:30041
  • 16. -, Shared values and normal families, preprint.
  • 17. Xuecheng Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc., to appear.
  • 18. P. J. Rippon and G. M. Stallard, Iteration of a class of hyperbolic meromorphic functions, Proc. Amer. Math. Soc. to appear. CMP 98:09
  • 19. J. Schiff, Normal Families, Springer, New York, Berlin, Heidelberg, 1993. MR 94f:30046
  • 20. W. Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 (1989), 241-289. MR 90k:30061
  • 21. N. Steinmetz, On the zeros of $(f^{(p)}+a_{p-1}f^{p-1}+\dots+a_0 f)f$, Analysis 7 (1987), 375-389. MR 89e:34059
  • 22. Yuefei Wang and Mingliang Fang, Picard values and normal families of meromorphic functions with multiple zeros, Acta Math. Sin., New Ser. 14 (1998), 17-26.
  • 23. L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813-817. MR 52:757
  • 24. -, Normal families: new perspectives, Bull. Amer. Math. Soc., New Ser. 35 (1998), 215-230. MR 99g:30048

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30D45, 30D30

Retrieve articles in all journals with MSC (1991): 30D45, 30D30

Additional Information

Walter Bergweiler
Affiliation: Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D–24098 Kiel, Germany

Received by editor(s): January 5, 1999
Received by editor(s) in revised form: March 9, 1999
Published electronically: June 21, 2000
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society