Uniqueness of entire functions sharing an entire function of smaller order with their derivatives (Q550581)

In 1996, \textit{R. Brück} [Result. Math. 30, No. 1--2, 21--24 (1996; Zbl 0861.30032)] conjectured that if \(f\) is a non-constant entire function satisfying \(\sigma_{2}(f) < \infty\), where the hyper-order \(\sigma_{2}(f)\) of \(f\) is not a positive integer, and if \(f\) and \(f'\) share one finite complex number \(a\) CM, then \(f - a = c(f' - a)\) for some constant \(c\neq 0\). This conjecture inspired a lot of researchers to work on the uniqueness problem of an entire function and its derivatives. The present paper also deals with such problems. A number of interesting results are proved in the paper, which involve the sharing of an entire function having slower growth than that of the function sharing it with derivatives.