Normal Families of Meromorphic Functions whose Derivatives Omit a Function

Abstract

Let $\cal F$ be a family of functions meromorphic on the plane domain D, and let h be a holomorphic function on D, h n= 0. Suppose that, for each $f \in {\cal F}$, f ^(m)(z) ≠ h(z) for z ∈ D. Then $t\cal F$ is normal on D (i) if all zeros of functions in $\cal F$ have multiplicity at least m + 3, or (ii) if all zeros of functions in $\cal F$ have multiplicity at least m + 2 and h has only multiple zeros on D, or (iii) if all poles of functions in $\cal F$ are multiple and all zeros have multiplicity at least m + 2.