Entire Functions and Their Derivatives Share Two Finite Sets

In this paper, we study the uniqueness of entire functions and prove the following theorem. Let n(>= 5), k be positive integers, and let S-1 = {z : z(n) - 1}, S-2 = {a(1), a(2), ... ,a(m)}, where a(1), a(2), ... , a(m), are distinct nonzero constants. If two non constant entire functions f and g satisfy E-f (S-1, 2) = E-g(S-1, 2) and E-f(k)(S-2, infinity) = E-g(k)(S-2, infinity), then one of the following cases must occur: (1) f = tg, {a(1), a(2), ... ,a(m)} = t{a(1), a(2), ... ,a(m)}, where t is a constant satisfying t(n) = 1; (2) f (z) = de(cz), g(z) = t/d e(-cz), {a(1), a(2), ... ,a(m)} = (-1)(k) c(2k) t{1/a(1), ... ,1/a(m)} where t, c, d are nonzero constants and t(n) = 1. The results in this paper improve the result given by Fang (M.L. Fang, Entire functions and their derivatives share two finite sets, Bull. Malaysian Math. Sc. Soc. 24(2001), 7-16).