Uniqueness theorems on difference monomials of entire functions
Summary: The aim of this paper is to discuss the uniqueness of the difference monomials \(f^n f(z + c)\). It is assumed that \(f\) and \(g\) are transcendental entire functions with finite order and \(E_{k)}(1, f^n f(z + c)) = E_{k)}(1, g^n g(z + c))\), where \(c\) is a nonzero complex constant and \(n, k\) are integers. It is proved that \(fg = t_1\) or \(f = t_2g\) for some constants \(t_2\) and \(t_3\) which satisfy \(t^{n+1}_2 = 1\) and \(t^{n+1}_3 = 1\), if \(k = 1\) and one of the following holds: (i) \(n \geq 6\) and \(k = 3\), (ii) \(n \geq 7\) and \(k = 2\), and (iii) \(n \geq 10\). It is an improvement of the result of \textit{X.-G. Qi}, \textit{L.-Z. Yang} and \textit{K. Liu} [Comput. Math. Appl. 60, No. 6, 1739--1746 (2010; Zbl 1202.30045)].
scientific article; zbMATH DE number 6078726