Abstract
Let \({\mathcal{F}}\) be a family of holomorphic functions defined in a domain \({\mathcal{D}}\) , let k( ≥ 2) be a positive integer, and let S = {a, b}, where a and b are two distinct finite complex numbers. If for each \({f \in \mathcal{F}}\) , all zeros of f(z) are of multiplicity at least k, and f and f (k) share the set S in \({\mathcal{D}}\) , then \({\mathcal{F}}\) is normal in \({\mathcal{D}}\). As an application, we prove a uniqueness theorem.
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References
Schiff J.: Normal Families. Springer, Berlin (1993)
Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Science Press, Beijing (1995, in Chinese); Kluwer Academic Publishers, The Netherlands (2003)
Schwick W.: Sharing values and normality. Arch Math. (Basel) 59, 50–54 (1992)
Fang M.L.: A note on sharing values and normality. J. Math. Study 29, 29–32 (1996)
Lü F., Xu J.F.: Sharing set and normal families of entire functions and their derivatives. Houston J. Math. 34, 1213–1223 (2008)
Li P.: Value sharing and differential equations. J. Math. Anal. Appl. 310, 412–423 (2005)
Pang X.C., Zalcman L.: Normal families and shared values. Bull. Lond. Math. Soc. 32, 325–331 (2000)
Gundersen G.G.: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. Lond. Math. Soc. 2(37), 88–104 (1988)
Yang L.Z.: Solution of a differential equation and its applications. Kodai Math. J. 22, 458–464 (1999)
Markushevich, A.I.: Theory of Functions of a Complex Variable, vol. 2. Prentice-Hall, Englewood Cliffs (1965). Translated by R. Silverman
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Li, Y. Sharing Set and Normal Families of Entire Functions. Results. Math. 63, 543–556 (2013). https://doi.org/10.1007/s00025-011-0216-8
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DOI: https://doi.org/10.1007/s00025-011-0216-8