Revista Matemática Iberoamericana


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Volume 11, Issue 2, 1995, pp. 355–373
DOI: 10.4171/RMI/176

On the singularities of the inverse to a meromorphic function of finite order

Walter Bergweiler[1] and Alexandre Eremenko[2]

(1) Christian-Albrechts-Universität zu Kiel, Germany
(2) Purdue University, West Lafayette, USA

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 $n≥1$ takes every finite non-zero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration theory of meromorphic functions.

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Bergweiler Walter, Eremenko Alexandre: On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoamericana 11 (1995), 355-373. doi: 10.4171/RMI/176