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.
CHANG JianMing Department of Mathematics, Changshu Institute of Technology, Changshu 215500, China
Let k, K ∈1N and F be a family of zero-free meromorphic functions in a domain D such that for each f ∈ F, f（k） - 1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most v=k/k＋1,where v is equal to the largest integer not exceeding k/k＋1.In particular, if K = k, then F is normal. The results are sharp.
