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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 ρ, then every asymptotic value of f, except at most 2ρ 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 n1 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:
30D30General theory of meromorphic functions
30D05Functional equations in the complex domain, iteration and composition of analytic functions