On the zeros of meromorphic functions of the form \(f(z)= \sum_{k=1}^ \infty {{a_ k} \over {z-z_ k}}\)
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.
scientific article; zbMATH DE number 569439