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

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 Brück conjecture are proved respectively by using Nevanlinna theory. As applications of these two results, the relationship between f and Δⁿ f (or between $f'$ and $f(z + 1)$) is established provided that f and Δⁿ f (or $f'$ and $f(z + 1)$) share a finite set. Moreover, some examples are provided to illustrate these results.