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

Citations:

Zbl 0088.28505; Zbl 0861.30032
Full Text:

References:

 [1] Yang, Uniqueness Theory of Meromorphic Functions (2003) · Zbl 1070.30011 · doi:10.1007/978-94-017-3626-8 [2] Laine, Nevanlinna Theory and Complex Differential Equations (1993) · doi:10.1515/9783110863147 [3] DOI: 10.1016/j.jmaa.2009.01.053 · Zbl 1180.30039 · doi:10.1016/j.jmaa.2009.01.053 [4] Hayman, Meromorphic Functions (1964) [5] Brück, Results in Math. 30 pp 21– (1996) · Zbl 0861.30032 · doi:10.1007/BF03322176 [6] DOI: 10.1016/j.jmaa.2005.04.010 · Zbl 1085.30026 · doi:10.1016/j.jmaa.2005.04.010 [7] DOI: 10.1112/jlms/s1-37.1.17 · Zbl 0104.29504 · doi:10.1112/jlms/s1-37.1.17 [8] DOI: 10.1007/s11139-007-9101-1 · Zbl 1152.30024 · doi:10.1007/s11139-007-9101-1 [9] DOI: 10.2307/1969890 · Zbl 0088.28505 · doi:10.2307/1969890
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