For f a meromorphic function on the plane domain D and a ∈ C, let Ēf(a) = {z ∈ D: f(z) = a}. Let F 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 ∈ F, E―f(0)=E―f(k)(b) and 0 < |f(k+1)(z)| ⩽ h whenever z ∈ Ēf(0), then F is a normal family on D. The case Ēf(0) = Ø is a celebrated result of Gu [5]. 1991 Mathematics Subject Classification 30D45, 30D35.

Abstract. Let 5 be a family of meromorphic functions on the unit disc 2~ and let a and b be distinct values. If for every fC3 e, f and fr share a and b on A, then 3 c is normal on A.

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Abstract. Let k ≥ 0 be an integer and α > 1. Let F be a family of functions meromorphic} is locally uniformly bounded away from zero, then F is normal.