Computational Methods and Function Theory 5 (2005), No. 1, 65--76
Copyright Heldermann Verlag 2005
Eleanor F. Clifford
e.clifford@lboro.ac.uk , Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
Let $a_0,\ldots,a_{k-1}$ be analytic functions on a domain $\Omega$. Let $\mathcal{F}$ be a family of meromorphic functions $f$ defined on $\Omega$ such that $f\neq0$ and $f^{(k)}+a_{k-1}f^{(k-1)}+\cdots+a_0f\neq0$ on $\Omega$, for all $f\in\mathcal{F}$. Then $\{f'/f:\,f\in\mathcal{F}\}$ is a normal family. Furthermore, let $a_0,\ldots,a_{k-1}$ be meromorphic functions on a domain $\Omega$. Let $\mathcal{F}$ be a family of meromorphic functions $f$ on $\Omega$ such that $f\neq0$, $f'\neq0$ and $f^{(k)}+a_{k-1}f^{(k-1)}+\cdots+a_0f\neq0$ on $\Omega$, for all $f\in\mathcal{F}$. Then $\{f'/f:\,f\in\mathcal{F}\}$ is a normal family. These two new criteria for normal families extend a recent result of Bergweiler and Langley, \cite[Corollary 1.1]{nonvan}.
Keywords: Normal families, meromorphic functions, Nevanlinna theory.
MSC 2000: 32A19, 32A20, 32A22.
[FullText-pdf (275 K)] [FullText-ps (465 K)]