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Normality concerning shared values

Authors:	JianMing Chang

Institution:	(1) Department of Mathematics, Changshu Institute of Technology, Changshu, 215500, China

Abstract:	Let $\mathcal{F}$ be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that $a/b \notin \mathbb{N}\backslash \{ 1\}$ . If for $f \in \mathcal{F}, f(z) = a \Rightarrow f'(z) = a$ and $f'(z) = b \Rightarrow f'(z) = b$ , then $\mathcal{F}$ is normal. We also construct a non-normal family $\mathcal{F}$ of meromorphic functions in the unit disk Δ={\|z\|<1} such that for every $f \in \mathcal{F}, f(z) = m + 1 \Leftrightarrow f'(z) = m + 1$ and $f'(z) = 1 \Leftrightarrow f'(z) = 1$ in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang. This work was supported by National Natural Science Foundation of China (Grant Nos. 10671093, 10871094) and the Natural Science Foundation of Universities of Jiangsu Province of China (Grant No. 08KJB110001), the Qing Lan Project of Jiangsu, China and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry

Keywords:	meromorphic function holomorphic functions normal family
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