Normal Families and Uniformly Discrete Exceptional Sets

Abstract

Let k be a positive integer, and ${\mathcal F}$ be a family of meromorphic functions in a domain D, all of whose zeros have multiplicity at least k + 2, and let h (≢ 0) be a holomorphic function. At each common zero of $f \in {\mathcal F}$ and h, the multiplicities m_f for f and m_h for h satisfy m_f≥ m_h + k + 1 for k < 1 and m_f ≥ 2m_h +3 for k = 1, and the exceptional sets {E_f} are locally uniformly discrete in D, where E_f = {z ∈ D: f (z)=0} ∪ {z ∈ D: f^(k)(z)= h(z)}. In this paper, the non-normal sequences of ${\mathcal F}$ are characterized, which shows the counterexample [1, Example 2, p. 49] is unique in some sense.