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

Abstract

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 $f^{\prime }{f}^n$ with $n\ge 1$ 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.