Abstract
Let \({\mathcal {F}}\) be a family of holomorphic functions on a domain D, and let \(h(\not \equiv 0)\) be a holomorphic function on D. If every \(f\in {\mathcal {F}}\), \(f(z)=0\Rightarrow |f'(z)|\le |h(z)|\), and \(f'(z)\ne h(z)\), then \({\mathcal {F}}\) is normal on D.
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References
Bergweiler, W., Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order. Rev Mat Iberoam 11, 355–373 (1995)
Bergweiler, W.: Normality and exceptional values of derivatives. Proc Amer Math Soc 129, 121–129 (2001)
Gu, Y.X.: A normal criterion of meromorphic families. Scientia, Math Issue I(13), 274–276 (1979)
Hayman, W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)
Liu, X.J., Ye, Y.S.: A criterion of normality concerning holomorphic functions whose derivative omits a function. Chin Ann Math 32B(5), 699–710 (2011)
Miranda, C.: Sur une nouveau critère de normalité pour les families de fonctions holomorphes. Bull Soc Math France 63, 185–196 (1935)
Pang, X.C., Fang, M.L., Zalcman, L.: Normal families of holomorphic functions with multiple zeros. Conform Geom Dyn 1, 101–106 (2007)
Pang, X.C., Zalcman, L.: Normal families and shared values. Bull London Math Soc 32, 325–331 (2000)
Rippon, P.J., Stallard, G.M.: Iteration of a class of hyperbolic meromorphic functions. Proc Amer Math Soc 127, 3251–3258 (1999)
Schiff, J.: Normal families. Springer, Berlin (1993)
Xu, Y.: Normal families and fixed-points of meromorphic functions. Monatsh Math 179, 471–485 (2016)
Yang, L.: Value Distribution Theory. Springer, Berlin (1993)
Yang, L.: Normality for families of meromorphic functions. Sci Sinica Ser A 29, 1263–1274 (1986)
Zhang, G.M., Pang, X.C., Zalcman, L.: Normal families and omitted functions II. Bull London Math Soc 41, 63–71 (2009)
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The authors are supported by NNSF of China(Grant No. 11471163) .
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Niu, P., Xu, Y. A normal criterion of families of holomorphic functions. Anal.Math.Phys. 11, 104 (2021). https://doi.org/10.1007/s13324-021-00539-8
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DOI: https://doi.org/10.1007/s13324-021-00539-8