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

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 >= 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.