×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Yang, Uniqueness Theory of Meromorphic Functions (2003) · Zbl 1070.30011
[2] Laine, Nevanlinna Theory and Complex Differential Equations (1993)
[3] DOI: 10.1016/j.jmaa.2009.01.053 · Zbl 1180.30039
[4] Hayman, Meromorphic Functions (1964)
[5] Brück, Results in Math. 30 pp 21– (1996) · Zbl 0861.30032
[6] DOI: 10.1016/j.jmaa.2005.04.010 · Zbl 1085.30026
[7] DOI: 10.1112/jlms/s1-37.1.17 · Zbl 0104.29504
[8] DOI: 10.1007/s11139-007-9101-1 · Zbl 1152.30024
[9] DOI: 10.2307/1969890 · Zbl 0088.28505
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.