Let k be a positive integer, let F be a family of zero-free meromorphic functions in a domain D, all of whose poles are multiple, and let h be a meromorphic function in D, all of whose poles are simple, h 0, ∞. If for each f ∈ F , f (k) (z) − h(z) has at most k zeros in D, ignoring multiplicities, then F is normal in D. The examples are provided to show that the result is sharp.