A generalization of Gu's normality criterion

Let F be a family of meromorphic functions on a domain D, k is an element of N and H be a normal family of meromorphic functions on D such that 0 is not in H and H has no sequence that converges to 0 or infinity spherically locally uniformly on D. If for every f is an element of F, f(z) not equal 0, and there exists an h(f) is an element of H such that f((k))(z) not equal h(f)(z) at every z is an element of D, then the family F is normal on D. This generalizes Gu's well-known normality criterion. It is interesting that the condition f(z) not equal 0 cannot be replaced by that all zeros of f have large multiplicities, at least k + 3 for instance.