A power of an entire function sharing one value with its derivative

In this paper, we investigate uniqueness problems of entire functions that share one value with one of their derivatives. Let ff be a non-constant entire function, nn and kk be positive integers. If fnfn and (fn)(k)(fn)(k) share 1 CM and n≥k+1n≥k+1, then fn=(fn)(k)fn=(fn)(k), and ff assumes the form f(z)=ceλnz, where cc is a non-zero constant and λk=1λk=1. This result shows that a conjecture given by Brück is true when F=fnF=fn, where n≥2n≥2 is an integer.