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A note on value distribution of difference polynomials. (English) Zbl 1201.30035
In [Ann. Math. (2) 70, 9–42 (1959; Zbl 0088.28505)] W. K. Hayman proved (among other results) that a differential polynomial f n +af ' -b with constant coefficients a,b admits infinitely many zeros, provided that f is transcendental entire and n3 (or n2 if b=0). The authors consider the difference counterpart of the expression above: f n (z)+f(z+c)-f(z)-b,n3 (or n2 if b=0). They prove that it has infinitely many zeros, provided that f is a transcendental entire function of finite order, not of period c. It is shown that one can replace b in this result with a nonzero function b(z), small compared to f. The authors also prove a result related to what can be called a difference counterpart of the R. Brück conjecture, see [Result. Math. 30, No.1–2, 21–24 (1996; Zbl 0861.30032)].

30D35Distribution of values (one complex variable); Nevanlinna theory
39B32Functional equations for complex functions