a1 Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D-24098 Kiel, Germany. e-mail: email@example.com
a2 School of Mathematical Sciences, University of Nottingham, NG7 2RD. e-mail: firstname.lastname@example.org
Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = Δf(z) = f(z + 1) − f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f. The results may be viewed as discrete analogues of existing theorems on the zeros of f' and f'/f.
(Received June 27 2005)
(Revised January 23 2006)
† Supported by the G.I.F., the German–Israeli Foundation for Scientific Research and Development, Grant G -809-234.6/2003.
‡ Supported by a grant from the Alexander von Humboldt Stiftung.