Entire functions that have the same fixed points with their first derivatives ✩

Let f be a nonconstant entire function, and let k ( 2) be an integer. We denote by Ef (z) = {z ∈ C: f (z) = z, counting multiplicities} the set consisting of all the fixed points of f . This paper proves that if f and f have the same fixed points, namely, Ef (z) = Ef (z), and if f (k)(z) = z whenever f (z) = z, then f ≡ f .  2003 Elsevier Inc. All rights reserved