Normal Criteria and Shared Values by Differential Polynomials

Jihong Wang, Qian Lu, Qilong Liao

.

DOI: 10.4236/apm.2011.14037                

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 f(z),g∈F , f'–afⁿand g'–agⁿ 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)f² and g'–a(z)g² 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.

Keywords

Normal Family, Meromorphic Function, Shared Value, Differential Polynomial

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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.