CRITERIA OF NORMALITY CONCERNING THE SEQUENCE OF OMITTED FUNCTIONS

• Chen, Qiaoyu (School of Statistics and Mathematics Shanghai Lixin University of Accounting and Finance) ;
• Qi, Jianming (Department of Mathematics and Physics Shanghai Dianji University)
• Published : 2016.09.30

Abstract

In this paper, we research the normality of sequences of meromorphic functions concerning the sequence of omitted functions. The main result is listed below. Let {$f_n(z)$} be a sequence of functions meromorphic in D, the multiplicities of whose poles and zeros are no less than k + 2, $k{\in}\mathbb{N}$. Let {$b_n(z)$} be a sequence of functions meromorphic in D, the multiplicities of whose poles are no less than k + 1, such that $b_n(z)\overset{\chi}{\Rightarrow}b(z)$, where $b(z({\neq}0)$ is meromorphic in D. If $f^{(k)}_n(z){\neq}b_n(z)$, then {$f_n(z)$} is normal in D. And we give some examples to indicate that there are essential differences between the normal family concerning the sequence of omitted functions and the normal family concerning the omitted function. Moreover, the conditions in our paper are best possible.

Acknowledgement

Supported by : National Natural Science Foundation of China, Natural Science Foundation of Shanghai, Chinese Postdoctoral Science Foundation

References

1. W. Bergweiler and W. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoam. 11 (1995), no. 2, 355-373.
2. Q. Y. Chen, X. C. Pang, and P. Yang, A new Picard type theorem concerning elliptic functions, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 17-30. https://doi.org/10.5186/aasfm.2015.4001
3. Q. Y. Chen, L. Yang, and X. C. Pang, Normal family and the sequence of omitted functions, Sci. China Math. 56 (2013), no. 9, 1821-1830. https://doi.org/10.1007/s11425-013-4580-6
4. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
5. S. Nevo, X. C. Pang, and L. Zalcman, Quasinormality and meromorphic functions with multiple zeros, J. Anal. Math. 101 (2007), 1-23. https://doi.org/10.1007/s11854-007-0001-5
6. X. C. Pang, S. Nevo, and L. Zalcman, Derivatives of meromorphic functions with multiple zeros and rational functions, Comput. Methods Funct. Theory 8 (2008), no. 1-2, 483-491. https://doi.org/10.1007/BF03321700
7. X. C. Pang, D. G. Yang, and L. Zalcman, Normal families and omitted functions, Indiana Univ. Math. J. 54 (2005), no. 1, 223-235. https://doi.org/10.1512/iumj.2005.54.2492
8. X. C. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc. 32 (2000), no. 3, 325-331. https://doi.org/10.1112/S002460939900644X
9. D. B. Tong, W. N. Zhou, and H. Wang, Exponential state estimation for stochastic complex dynamical networks with multi-delayed base on adaptive control, Int. J. Control, Autom. Syst. 12 (2014), no. 5, 963-968. https://doi.org/10.1007/s12555-013-0323-2
10. D. B. Tong, W. N. Zhou, X. G. Zhou, J. Yang, L. Zhang, and Y. Xu, Exponential synchronization for stochastic neural networks with multi-delayed and Markovian switching via adaptive feedback control, Commun. Nonlinear Sci. Numer. Simul. 29 (2015), no. 1-3, 359-371. https://doi.org/10.1016/j.cnsns.2015.05.011
11. Y. F. Wang and M. L. Fang, Picard values and normal families of meromorphic functions with multiple zeros, Acta Math. Sinica (N.S.) 14 (1998), no. 1, 17-26. https://doi.org/10.1007/BF02563879
12. Y. Xu, Picard values and derivatives of meromorphic functions, Kodai Math. J. 28 (2005), no. 1, 99-105. https://doi.org/10.2996/kmj/1111588039
13. Y. Xu, Normal families and exceptional functions, J. Math. Anal. Appl. 329 (2007), no. 2, 1343-1354. https://doi.org/10.1016/j.jmaa.2006.07.021
14. L. Yang, Value Distribution Theory, Springer, Berlin, 1993.
15. P. Yang, A quasinormal criterion of meromorphic functions and its application, J. Inequal. Appl. 2014 (2014), 1-24. https://doi.org/10.1186/1029-242X-2014-1
16. P. Yang and X. J. Liu, On the kth derivatives of meromorphic functions and rational functions, J. East China. Norm. Univ. Natur. Sci. Ed. 2014 (2014), no. 4, 8-17.
17. P. Yang and S. Nevo, Derivatives of meromorphic functions with multiple zeros and elliptic functions, Acta Math. Sin. 29 (2013), no. 7, 1257-1278. https://doi.org/10.1007/s10114-013-2375-x
18. G. M. Zhang, X. C. Pang, and L. Zalcman, Normal families and omitted functions. II, Bull. London Math. Soc. 41 (2009), no. 1, 63-71. https://doi.org/10.1112/blms/bdn103