## 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.

### MSC:

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

Zbl 0797.31002
Full Text:

### References:

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