Meromorphic functions of the form \(f(z)=\sum^ \infty_{n=1}a_ n/(z-z_ n)\).
This is an important paper containing several results on the zeros of transcendental meromorphic functions of the form \(f(z)=\sum_{n=1}^\infty\frac{a_n}{z-z_n}\) with complex coefficients \(a_n\) and such that \(\sum_{z_n\neq0}| a_n/z_n| <\infty\). The methods applied here include quasiconformal surgery, Fourier series, and Baernstein's spread theorem. Some examples of the results obtained are as follows: (1) Given \(f\) of the above type and of finite order such that \(a_n>0\) are real, \(\sum_{n=1}^\infty a_n=\infty\), and that either \(\liminf_{r\to\infty}\frac{T(r,f)}f<\infty\), or \(| z-\overline z_n| \leq| z_n| ^{1-\varepsilon}\) for all \(n\) large enough, then \(f(z)-S(z)\) has infinitely many zeros, provided \(S(z)\) is rational. This result establishes a conjecture from [J. Lond. Math. Soc. II. Ser. 47, 309--320 (1993; Zbl 0797.31002)] in a special case. (2) The authors pose a conjecture that \(f\) of the above type has infinitely many zeros, provided \(\sup\{\,| \arg a_n| ;n\in {\mathbb{N}}\,\}<\pi/2\), true if \(f\) is of finite lower order and \(\sum| a_n| <\infty\). Two theorems, too technical to be repeated here, are related to this conjecture in the case of \(\rho(f)\leq1/2\), resp. of \(\rho(f)<1\). These results then may be utilized to the fixpoint theory in (3): Given \(F\) transcendental and meromorphic of order \(\leq1/2\) and with finitely many poles, and given \(0<c<1\), then \(F\) has infinitely many fixpoints \(z\) such that \(| F'(z)| \geq c\). Some examples will also be offered to illuminate the results in this paper.
scientific article; zbMATH DE number 2104148