The difference analogues and delay-differential analogues of the Brück conjecture and their application

Abstract: Let ( f ) be a transcendental entire function with hyper-order strictly less than 1. Under certain conditions, the difference analogues and delay-differential analogues of the Br\"uck conjecture are proved respectively by using Nevanlinna theory. As applications of these two results, the relationship between $f$ and $\Delta^n f$ (or between $f'$ and $f(z + 1)$) is established provided that $f$ and $\Delta^n f$ (or $f'$ and $f(z + 1)$) share a finite set. Moreover, some examples are provided to illustrate these results.