Meromorphic functions of the form \(f(z)=\sum^ \infty_{n=1}a_ n/(z-z_ n)\). (English) Zbl 1061.30022

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.


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory


Zbl 0797.31002
Full Text: DOI EuDML


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