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Value distribution and shared sets of differences of meromorphic functions. (English) Zbl 1188.30044
Summary: We investigate value distribution and uniqueness problems for difference polynomials of meromorphic functions. In particular, we show that, for a finite order transcendental meromorphic function f with λ(1/f)<ρ(f) and a non-zero complex constant c, if n2 then f(z) n f(z+c) assumes every non-zero value a infinitely often. This research also shows that there exist two sets S 1 with 9 (resp. 5) elements and S 2 with 1 element such that, for a finite order non-constant meromorphic (resp. entire) function f and a non-zero complex constant c, E f(z) (S j )=E f(z+c) (S j ) (j=1,2) implies f(z)f(z+c). This gives an answer to a question of F. Gross [Complex Anal., Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 51–67 (1977; Zbl 0357.30007)] concerning a finite order meromorphic function f and its shift.

30D35Distribution of values (one complex variable); Nevanlinna theory
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