For f a meromorphic function on the plane domain D and a ∈ [Copf], let Ēf(a) = {z ∈ D[ratio ]f(z) = a}. Let [Fscr] be a family of meromorphic functions on D, all of whose zeros are of multiplicity at least k. If there exist b ≠ 0 and h > 0 such that for every f ∈ [Fscr], Ēf(0) = Ēf(k)(b) and 0 < [mid ]f(k+1)(z)[mid ] [les ] h whenever z ∈ Ēf(0), then [Fscr] is a normal family on D. The case Ēf(0) = Ø is a celebrated result of Gu [5].