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

Walter Bergweiler; Alexander Eremenko

Revista Matemática Iberoamericana (1995)

  • Volume: 11, Issue: 2, page 355-373
  • ISSN: 0213-2230

Abstract

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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'fn 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.

How to cite

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Bergweiler, Walter, and Eremenko, Alexander. "On the singularities of the inverse to a meromorphic function of finite order.." Revista Matemática Iberoamericana 11.2 (1995): 355-373. <http://eudml.org/doc/39474>.

@article{Bergweiler1995,
abstract = {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'fn 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.},
author = {Bergweiler, Walter, Eremenko, Alexander},
journal = {Revista Matemática Iberoamericana},
keywords = {Función meromorfa; Superficies Riemann; Ley límite; Teoría de singularidades},
language = {eng},
number = {2},
pages = {355-373},
title = {On the singularities of the inverse to a meromorphic function of finite order.},
url = {http://eudml.org/doc/39474},
volume = {11},
year = {1995},
}

TY - JOUR
AU - Bergweiler, Walter
AU - Eremenko, Alexander
TI - On the singularities of the inverse to a meromorphic function of finite order.
JO - Revista Matemática Iberoamericana
PY - 1995
VL - 11
IS - 2
SP - 355
EP - 373
AB - 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'fn 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.
LA - eng
KW - Función meromorfa; Superficies Riemann; Ley límite; Teoría de singularidades
UR - http://eudml.org/doc/39474
ER -

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