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Zbl 0830.30016
Bergweiler, Walter; Eremenko, Alexandre
On the singularities of the inverse to a meromorphic function of finite order. (English)
[J] Rev. Mat. Iberoam. 11, No.2, 355-373 (1995). ISSN 0213-2230

Summary: Our main result implies the following theorem: Let $f$ be a transcendental meromorphic function in the complex plane. If $f$ has finite order $\rho$, then every asymptotic value of $f$, except at most $2 \rho$ of them, is a limit point of critical values of $f$. We give several applications of this theorem. For example we prove that if $f$ is a transcendental meromorphic function then $f'f^n$ with $n \ge 1$ takes every finite nonzero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration theory of meromorphic functions.

MSC 2000:: *30D30 General theory of meromorphic functions
30D05 Functional equations in the complex domain

Cited in: Zbl 1079.30041 Zbl 1160.30341 Zbl 1045.30015 Zbl 1046.30014 Zbl 0830.30019

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