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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 $$n\geq 3$$ (or $$n\geq 2$$ if $$b=0$$). The authors consider the difference counterpart of the expression above: $$f^n(z)+f(z+c)-f(z)-b, n\geq 3$$ (or $$n\geq 2$$ 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)].

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 39B32 Functional equations for complex functions
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##### References:
  Yang, Uniqueness Theory of Meromorphic Functions (2003) · Zbl 1070.30011  Laine, Nevanlinna Theory and Complex Differential Equations (1993)  DOI: 10.1016/j.jmaa.2009.01.053 · Zbl 1180.30039  Hayman, Meromorphic Functions (1964)  Brück, Results in Math. 30 pp 21– (1996) · Zbl 0861.30032  DOI: 10.1016/j.jmaa.2005.04.010 · Zbl 1085.30026  DOI: 10.1112/jlms/s1-37.1.17 · Zbl 0104.29504  DOI: 10.1007/s11139-007-9101-1 · Zbl 1152.30024  DOI: 10.2307/1969890 · Zbl 0088.28505
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