Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

We study the normality of families of meromorphic functions defined in terms of certain omitted functions. In particular, we prove the following results. Firstly, if is a family of meromorphic functions in a domain D ⊂ ℂ, and a(z), b(z) and c(z) are distinct meromorphic functions in D and if, for all f ∈ and all z ∈ D, f(z) ≠ a(z), f(z) ≠ b(z) and f(z) ≠ c(z), then is normal in D. Secondly, letting R(w) be a rational function of degree greater than or equal to 3 and be a family of functions meromorphic in a domain D ⊂ ℂ, if there exists a non-constant meromorphic function α(z) in D such that, for all f ∈ and z ∈ D, R(f(z)) ≠ α(z), then is normal in D.

Let k be a positive integer and b a nonzero constant. Suppose that F is a family of meromorphic functions in a domain D. If each function f ∈ F has only zeros of multiplicity at least k + 2 and for any two functions f, g ∈ F, f and g share 0 in D and f(k) and g(k) share b in D, then F is normal in D. The case f ≠ 0, f(k) ≠ b is a celebrated result of Gu.

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].