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.
CHANG JianMing Department of Mathematics, Changshu Institute of Technology, Changshu 215500, China
Let k, K ∈ N 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 ν = K k+1 , where ν is equal to the largest integer not exceeding K/k+1 . In particular, if K = k, then F is normal. The results are sharp.
