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

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.


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
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