On the zeros of meromorphic functions of the form \(f(z)= \sum_{k=1}^ \infty {{a_ k} \over {z-z_ k}}\)
type: Item
instance of: scholarly article
title: On the zeros of meromorphic functions of the form \(f(z)= \sum_{k=1}^ \infty {{a_ k} \over {z-z_ k}}\)
zbMATH Open document ID: 0818.30020
DOI: 10.1007/BF02835958
author: John Rossi
author: Alexandre Eremenko
author: James K. Langley
published in: Journal d'Analyse Mathématique
publication date: 1995-08-06T00:00:00Z
review text: Let \(\{z_ k\}\) be a sequence of complex numbers and \(\{a_ k\}\) a sequence of positive real numbers. The authors study functions \(f\) defined by \[ f(z)= \sum^ \infty_{k= 1} (a_ k)/(z- z_ k)\leqno{(*)} \] and obtain some results which complement those in a paper by \textit{J. Clunie}, \textit{A. Eremenko} and \textit{J. Rossi} [J. Lond. Math. Soc. (to appear)]. In particular, if \(f\) has order at most one, minimal type, then \(f\) has an infinite set of zeros. Further, if \(f\) defined by \((*)\) has the \(a_ k\) as integers for which \(a_ k\geq -1\) for \(k\geq k_ 0\), then \(f\) has infinitely many zeros. This latter theorem is equivalent to the result that \(F'\) has infinitely many zeros when \(F\) is a transcendental meromorphic function of order at most one, minimal type (answering a question of W. K. Hayman). Additional results on functions of the form \((*)\) show that if \(F\) is an entire function with order less than one for which the multiplicity of the zeros is uniformly bounded, then \(\delta(0, F'/F)< 1\). Proofs of the theorems rely on the understanding of a large number and diverse sets of known results on meromorphic functions along with considerable technical expertise.
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 569439
reviewed by: Linda R. Sons
MaRDI profile type: Publication
cites work: The Size of the Set on Which a Meromorphic Function is Large
cites work: Q4182997
cites work: On Equilibrium Points of Logarithmic and Newtonian Potentials
cites work: Convolution inequalities, regular variation and exceptional sets
cites work: Minimum of the modulus of an entire function on the sequence of Pólya peaks
cites work: Q3250731
cites work: Q5723337
cites work: Q3852455
cites work: Q4692700
cites work: On the Growth of Solutions of f � � + gf � + hf = 0
cites work: On the growth of meromorphic functions having at least one deficient value
cites work: Q3259806
cites work: A theorem on Nevanlinna deficiencies
description: scientific article; zbMATH DE number 569439