Computational Methods and Function Theory 4 (2004), No. 2, 327--339
Copyright Heldermann Verlag 2004
Eleanor F. Clifford
eleanor.clifford@maths.nottingham.ac.uk , Department of Mathematics, University of Nottingham, Nottingham, NG7 2RD, U.K.
We consider the differential operator $\Lambda_k$ defined by $$ \Lambda_k(y)= \Psi_k(y)+a_{k-1}\Psi_{k-1}(y)+\cdots+a_1\Psi_1(y)+a_0, $$ where $a_0,\ldots,a_{k-1}$ are analytic functions of restricted growth and $\Psi_k(y)$ is a differential operator defined by $\Psi_1(y)=y$ and $\Psi_{k+1}(y)=y\Psi_k(y)+(\Psi_k(y))'$ for $k\in\mathbb{N}$. We suppose that $k\geq3$, that $F$ is a meromorphic function on an annulus $\mathcal{A}(r_0)$, and that $\Lambda_k(F)$ has all its zeros on a set $E$ such that $E$ has no limit point in $\mathcal{A}(r_0)$. We suppose also that all simple poles $a$ of $F$ in $\mathcal{A}(r_0)\setminus E$ have $\res(F,a)\notin\{1,\ldots,k-1\}$. We then deduce that $F$ is a function of restricted growth in the Nevanlinna sense. This extends a theorem of Bergweiler and Langley \cite{nonvan}. We show also that this result does not hold when $a_0,\ldots,a_{k-1}$ are meromorphic functions.
Keywords: Meromorphic functions, Nevanlinna theory.
MSC 2000: 30D30, 30D35.
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