Abstract
Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f (k) share 0, and |f(z)| ≥ M whenever f (k)(z) = h(z), then F is normal in D. The condition that f and f (k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f (k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f (k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.
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References
H. H. Chen and M. L. Fang, Shared values and normal families of meromorphic functions, Journal of Mathematical Analysis and Applications 260 (2001), 124–132.
M. L. Fang and L. Zalcman, Normal families and shared values of meromorphic functions, Annales Polonici Mathematici 80 (2003), 133–141.
Y. X. Gu, Un critère de normalité de fonctions méromorphes, Sci. Sinica, Special Issue 1 (1979), 267–274.
W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
X. C. Pang, M. L. Fang and L. Zalcman, Normal families of holomorphic functions with multiple zeros, Conformal Geometry and Dynamics 11 (2007), 101–106.
X. C. Pang, D. G. Yang and L. Zalcman, Normal families of meromorphic functions whose derivatives omit a function, Computational Methods and Function Theory 2 (2002), 257–265.
X. C. Pang and L. Zalcman, Normal families and shared values, The Bulletin of the London Mathematical Society 32 (2000), 325–331.
J. Schiff, Normal Families, Springer-Verlag, New York, 1993.
Y. F. Wang and M. L. Fang, Picard values and normal families of meromorphic functions with multiple zeros, Acta Mathematica Sinica (N.S.) (1) 14 (1998), 17–26.
L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993.
G. M. Zhang, X. C. Pang and L. Zalcman, Normal families and omitted functions II, The Bulletin of the London Mathematical Society 41 (2009), 63–71.
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Project supported by the NNSF of China (Grant No. 10771076) and the NSF of Guangdong Province, China (Grant No. 07006700).
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Chen, JF., Fang, ML. Normal families and shared functions of meromorphic functions. Isr. J. Math. 180, 129–142 (2010). https://doi.org/10.1007/s11856-010-0097-7
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DOI: https://doi.org/10.1007/s11856-010-0097-7