On the singularities of the inverse to a meromorphic function of finite order. (English) Zbl 0830.30016

Bergweiler, Walter; Eremenko, Alexandre

Rev. Mat. Iberoam. 11, No. 2, 355-373 (1995).

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 \(n \geq 1\) 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:

30D30	Meromorphic functions of one complex variable (general theory)
30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable