## 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

### Keywords:

normal family; meromorphic functions; shared values

Zbl 1048.30018
Full Text:

### 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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