Uniqueness and value-sharing of entire functions

In this paper, we study the uniqueness of entire functions and prove the following theorem. Let f(z) and g(z) be two nonconstant entire functions, n, k two positive integers with n > 2k + 4. If [fn(z)](k) and [gn(z)](k) share 1 with counting the multiplicity, then either f(z) = c1ecz, g(z) = c2e−cz, where c1, c2, and c are three constants satisfying (−1)k(c1c2)n(nc)2k = 1, or f(z) ≡ tg(z) for a constant t such that tn = 1.