Normal Criteria and Shared Values by Differential Polynomials

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*'–*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.

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

Cite this paper

nullJ. 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.

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

References

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[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.

[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.