J. Aust. Math. Soc.  78 (2005), 37-57
Multiplicities in Hayman's alternative

 Walter Bergweiler   Mathematisches Seminar   Christian-Albrechts-Universität zu Kiel   Ludewig-Meyn Str. 4   D-24098 Kiel   Germany  bergweiler@math.uni-kiel.de
 and
 J. K. Langley   School of Mathematical Sciences   University of Nottingham   NG7 2RD   UK  jkl@maths.nott.ac.uk

Abstract
In 1959 Hayman proved an inequality from which it follows that if is transcendental and meromorphic in the plane then either takes every finite complex value infinitely often or each derivative , , takes every finite non-zero value infinitely often. We investigate the extent to which these values may be ramified, and we establish a generalization of Hayman's inequality in which multiplicities are not taken into account.
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