Computational Methods and Function Theory 2 (2002), No. 2, 481--508
Copyright Heldermann Verlag 2002
Jürgen Grahl
grahl@mathematik.uni-wuerzburg.de, Mathematisches Institut der Universität Würzburg, Am Hubland, 97074 Würzburg, Germany.
We show that if $\mathcal{F}$ is a family of non-vanishing meromorphic functions in the unit disk $\mathbb{D}$ with $P[f](z)\ne 1$ for all $z\in\mathbb{D}$ and all $f\in\mathcal{F}$ where $P$ is a differential polynomial satisfying certain conditions, then $\mathcal{F}$ is normal. This generalizes former results of W.\ Schwick \cite{schwick} and of M.-L.\ Fang \cite{fang0}. Furthermore, we give the corresponding Picard type theorems generalizing Hayman's Alternative.
Keywords: Normal families, differential polynomials, Hayman's Alternative, Zalcman's Lemma, Bloch's Principle.
MSC 2000: 30D35, 30D45, 34M05.
[FullText-pdf (408 K)] [FullText-ps (380 K)]