Normality concerning shared values  

作　　者：CHANG JianMing Department of Mathematics, Changshu Institute of Technology, Changshu 215500, China 

出　　处：《Science China Mathematics》2009年第8期1717-1722,共6页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 10671093, 10871094);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

摘　　要：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.Let 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 ∈ N \ {1}. If for every f ∈ F, f(z) = a =■ f (z) = a and f (z) = b =■ f (z) = b, then F is normal. We also construct a non-normal family F of meromorphic functions in the unit disk Δ = {|z| < 1} such that for every f ∈ F, f(z) = m + 1 f (z) = m + 1 and f (z) = 1 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.

关 键 词：meromorphic function holomorphic functions normal family 30D45 

分 类 号：O174.52[理学—数学]