Normal families and multiple values

Abstract.

Let ${{\mathcal{F}}}$ be a family of meromorphic functions in a domain D, and let k be a positive integer, and let b be a nonzero complex number. If, for each $f \in {{\mathcal{F}}}$, f ≠ 0, $f^{(k)} \neq 0$ and the zeros of $f^{(k)} - b$ have multiplicity at least 3 for k = 1 and 2 for k ≥ 2, then ${{\mathcal{F}}}$ is normal in D. Examples show that the multiplicity of the zeros of $f^{(k)} - b$ is best possible.