Normality and quasinormality of zero-free meromorphic functions

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 $\nu = \left {\tfrac{K} {{k + 1}}} \right] $\nu = \left {\tfrac{K} {{k + 1}}} \right] , where ν is equal to the largest integer not exceeding $\tfrac{K} {{k + 1}} $\tfrac{K} {{k + 1}} . In particular, if K = k, then F is normal. The results are sharp.