JournalsrmiVol. 11, No. 2pp. 355–373

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

  • Walter Bergweiler

    Christian-Albrechts-Universität zu Kiel, Germany
  • Alexandre Eremenko

    Purdue University, West Lafayette, USA
On the singularities of the inverse to a meromorphic function of finite order cover
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Abstract

Our main result implies the following theorem: Let ff be a transcendental meromorphic function in the complex plane. If ff has finite order ρ\rho,  then every asymptotic value of ff, except at most 2ρ2\rho of them, is a limit point of critical values of ff.

 We give several applications of this theorem. For example we prove that if ff is a transcendental meromorphic function then ffnf'f^n with n1n≥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.

Cite this article

Walter Bergweiler, Alexandre Eremenko, On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoam. 11 (1995), no. 2, pp. 355–373

DOI 10.4171/RMI/176