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)].


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39B32 Functional equations for complex functions
Full Text: DOI


[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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