Scientific Research

An Academic Publisher

**Normal Criteria and Shared Values by Differential Polynomials** ()

For a family of meromorphic functions on a domain D, it is discussed whether F is normal on D if for every pair functions

*f*(z),*g*∈*F*,*f*'–*af*^{n}and*g*'–*ag*^{n}share value*d*on D when n=2,3, where*a, b*are two complex numbers,*a*≠0,∞,*b*≠∞.Finally, the following result is obtained:Let*F*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*a*(z) not idendtically equal to zero,*d*(z) analytic in D, such that for each pair of functions*f*and in*F*,*f*'–*a*(z)*f*^{2}and*g*'–*a*(z)*g*^{2}share the function*d*(z) . If*a*(z) has only a multiple zeros and*f*(z)≠∞ whenever*a*(z)=0 , then*F*is normal in D.Share and Cite:

J. Wang, Q. Lu and Q. Liao, "Normal Criteria and Shared Values by Differential Polynomials,"

*Advances in Pure Mathematics*, Vol. 1 No. 4, 2011, pp. 210-217. doi: 10.4236/apm.2011.14037.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | 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. |

[2] | 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. |

[3] | 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. |

[4] | W. K. Hayman, “Picard Values of Meromorphic Functions and the Its Derivatives,” Annals of Mathematics, Vol. 70, 1959, pp. 9-42. |

[5] | W. K. Hayman, “Meromorphic Functions,” Clar-endon, Oxford, 1964. |

[6] | 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. |

[7] | X. J. Huang and Y. X. Gu, “Normal Families of Meromorphic Functions,” Results in Mathematics, Vol. 49, 2006, pp. 279-288. |

[8] | S. Y. Li, “On Normal Criterion of Meromorphic Functions,” Journal of Fu-jian Normal University, Vol. 25 1984, pp. 156-158. |

[9] | X. J. Li, “Proof of Hayman’s Conjecture on Normal Families,” Sci-ence in China Series A, Vol. 28, 1985, pp. 596-603. |

[10] | X. C. Pang, “On Normal Criterion of Meromorphic Functions,” Sci-ence in China Series A, Vol. 33, No. 5, 1990, pp. 521-527. |

[11] | 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. |

[12] | 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. |

[13] | L. Zalcman, “Normal Families: New Perspectives,” Bulletin (New Series) of the American Mathematical Society, Vol. 35, No. 3, 1998, pp. 215-230. |

[14] | 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. |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.