## 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.

### 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
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