Normal Criteria and Shared Values by Differential Polynomials
- 1
- 2
- 3
Abstract
For a family of meromorphic functions on a domain D, it is discussed whether F is normal on D if for every pair functions <i>f</i>(z),<i>g</i>∈<i>F</i> , <i>f</i>'–<i>af</i><sup>n</sup>and <i>g</i>'–<i>ag</i><sup>n</sup> share value <i>d</i> on D when n=2,3, where <i>a, b</i> are two complex numbers, <i>a</i>≠0,∞,<i>b</i>≠∞.Finally, the following result is obtained:Let <i>F</i> be a family of meromorphic functions in D, all of whose poles have multiplicity at least 4 , all of whose zeros have multiplicity at least 2. Suppose that there exist two functions <i>a</i>(z) not idendtically equal to zero, <i>d</i>(z) analytic in D, such that for each pair of functions <i>f</i> and in <i>F</i> , <i>f</i>'–<i>a</i>(z)<i>f</i><sup>2</sup> and <i>g</i>'–<i>a</i>(z)<i>g</i><sup>2</sup> share the function <i>d</i>(z) . If <i>a</i>(z) has only a multiple zeros and <i>f</i>(z)≠∞ whenever <i>a</i>(z)=0 , then <i>F</i> is normal in D.
- W. Bergweiler and A. Eremenko, “On the Singularities of the Inverse to a Meromorphic Function of Finite Order,” Revista Matemática Iberoamericana, Vol. 11, No. 2, 1995, pp. 355-373.
- H. H. Chen and M. L. Fang, “On the Value Dis-tribution of ,” Science in China Series A, Vol. 38, No. 7, 1995, pp. 789-798.
- M. L. Fang and W. J. Yuan, “On the Normality for Families of Meromorphic Functions,” Indian Journal of Mathematics, Vol. 43, 2001, pp. 341-350.
- W. K. Hayman, “Picard Values of Meromorphic Functions and the Its Derivatives,” Annals of Mathematics, Vol. 70, 1959, pp. 9-42.
- W. K. Hayman, “Meromorphic Functions,” Clar-endon, Oxford, 1964.
- X. J. Huang and Y. X. Gu, “Normal Families of Meromorphic Functions with Multiple Zeros and Poles,” Journal of Mathematical Analysis and Applications, Vol. 295, No. 2, 2004, pp. 611-619.
- X. J. Huang and Y. X. Gu, “Normal Families of Meromorphic Functions,” Results in Mathematics, Vol. 49, 2006, pp. 279-288.
- S. Y. Li, “On Normal Criterion of Meromorphic Functions,” Journal of Fu-jian Normal University, Vol. 25 1984, pp. 156-158.
- X. J. Li, “Proof of Hayman’s Conjecture on Normal Families,” Sci-ence in China Series A, Vol. 28, 1985, pp. 596-603.
- X. C. Pang, “On Normal Criterion of Meromorphic Functions,” Sci-ence in China Series A, Vol. 33, No. 5, 1990, pp. 521-527.
- X. C. Pang, D. G. Yang, and L. Zalcman, “Normal Families of Meromorphic Functions Omitting a Func-tion ii,” Computational Methods and Function Theory, Vol. 2, No. 1, 2002, pp. 257-265.
- Y. F. Wang and M. L. Fang, “Picard Values and Normal Families of Meromorphic Func-tions with Multiple Zeros,” Acta Mathematica Sinica, Chinese Series, Vol. 41, No. 4, 1998, pp. 743-748.
- L. Zalcman, “Normal Families: New Perspectives,” Bulletin (New Series) of the American Mathematical Society, Vol. 35, No. 3, 1998, pp. 215-230.
- Q. C. Zhang, “Normal Families of Meromorphic Functions Concerning Sharing Values,” Journal of Mathe-matical Analysis and Applications, Vol. 338, No. 1, 2008, pp. 545-551.