Communications of the
Korean Mathematical Society CKMS

In this paper, we utilize the Nevanlinna theory and uniqueness theory of meromorphic function to investigate the differential-diff\-erence analogue of Br\"{u}ck conjecture. In other words, we consider \linebreak $\Delta_{\eta}f(z)=f(z+\eta)-f(z)$ and $f'(z)$ share one value or one small function, and then obtain the precise expression of transcendental entire function $f(z)$ under certain conditions, where $\eta\in \mathbb{C}\setminus \{0\}$ is a constant such that $f(z+\eta)-f(z)\not\equiv 0$.

Keywords: Nevanlinna theory, uniqueness theory, Br\"{u}ck conjecture, differential-difference equation

MSC numbers: 39B32, 34M05, 30D35