Normal families of meromorphic functions with multiple zeros and poles

Let k be a positive integer with k greater than or equal to 2 and let F be a family of functions meromorphic on a domain D in C, all of whose poles have multiplicity at least 3, and of whose zeros all have multiplicity at least k + 1. Let a(z) be a function holomorphic on D, a(z) not equivalent to 0. Suppose that for each f is an element of F, f ((k)) (z) not equal a(z) for z is an element of D. Then F is a normal family on D. (C) 2004 Elsevier Inc. All rights reserved.