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Normal families and shared values of meromorphic functions. (English) Zbl 1179.30033

Let \(f\) and \(g\) be meromorphic functions on a domain \(D\), and let \(a\) and \(b\) be two complex numbers. If \(g(z)=b\) whenever \(f(z)=a\), we write \(f(z)=a\Rightarrow g(z)=b\). If \(f(z)=a\Rightarrow g(z)=b\) and \(g(z)=b\Rightarrow f(z)=a\), we write \(f(z)=a\Leftrightarrow g(z)=b\).
The authors of this paper prove the following theorem which improves a result of M. Fang and L. Zalcman [Comput. Methods Funct. Theory 2, No. 2, 385–395 (2002; Zbl 1048.30018)].
Let \(\mathcal F\) be a family of meromorphic functions in a domain \(D\), let \(q, k\) be two positive integers, and let \(a, b\) be two non-zero complex numbers. If, for each \(f\in \mathcal F\), the zeros of \(f\) have multiplicity at least \(k+1\), and \(f=a\Leftrightarrow G(f)=b\), where \(G(f)=P(f^{(k)})+H(f)\) is a differential polynormial of \(f\) satisfying \(q\geq \gamma_H\) and \(\frac{\Gamma}{\gamma}|_{H}<k+1\), then \(\mathcal F\) is normal in \(D\).

MSC:

30D45 Normal functions of one complex variable, normal families

Citations:

Zbl 1048.30018
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References:

[1] W. Schwick, Sharing values and normality, Arch. Math. (Basel) 59 (1992), no. 1, 50-54. · Zbl 0758.30028 · doi:10.1007/BF01199014
[2] M. Fang, A note on sharing values and normality, J. Math. Study 29 (1996), no. 4, 29-32. · Zbl 0919.30027
[3] M. Fang and L. Zalcman, Normal families and shared values of meromorphic functions. III, Comput. Methods Funct. Theory 2 (2002), no. 2, 385-395. · Zbl 1048.30018 · doi:10.1007/BF03321856
[4] X. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc. 32 (2000), no. 3, 325-331. · Zbl 1030.30031 · doi:10.1112/S002460939900644X
[5] Y. Wang and M. Fang, Picard values and normal families of meromorphic functions with multiple zeros, Acta Math. Sinica (N.S.) 14 (1998), no. 1, 17-26. · Zbl 0909.30025 · doi:10.1007/BF02563879
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