Normality and quasinormality of zero-free meromorphic functions

Let k, K . . 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 .=[Kk+1] , where . is equal to the largest integer not exceeding Kk+1. In particular, if K = k, then F is normal. The results are sharp.