Value distribution of the difference operator
This paper is devoted to considering value distribution of difference operators of transcendental entire functions. The key result is the following theorem: Let \(f\) be a transcendental entire function of finite order, \(p\) a non-vanishing polynomial, \(c\) a non-zero complex number, and \(n\geq 2\). If \(\Delta_{c}f:=f(z+c)-f(z)\) is not vanishing identically, then \(f(z)^{n}\Delta_{c}f-p(z)\) has infinitely many zeros. It will be shown by simple examples that the assertion fails if \(f\) is of infinite order, resp. if \(n=1\). This theorem is an extension to a previous result due to \textit{I. Laine} and \textit{C.-C. Yang} in [Proc. Japan Acad., Ser. A 83, 148--151 (2007; Zbl 1153.30030)] .
scientific article; zbMATH DE number 5565241