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 f be a non-constant entire function, n and k be positive integers. If f n and ( f n ) ( k ) share 1 CM and n ≥ k + 1 , then f n = ( f n ) ( k ) , and f assumes the form f ( z ) = c e λ n z , where c is a non-zero constant and λ k = 1 . This result shows that a conjecture given by Brück is true when F = f n , where n ≥ 2 is an integer.