Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

We use the theory of normal families to prove the following. Let Q(1)(z) = a(1)z(p) + a(1),(p-1)z(p-1) + ... + a(1,0) and Q2(z) = a(2)z(p) + a(2,p- 1)z(p-1) + ... + a(2,0) be two polynomials such that deg Q(1) = deg Q(2) = p (where p is a nonnegative integer) and a(1), a(2)(a2 not equal 0) are two distinct complex numbers. Let f(z) be a transcendental entire function. If f(z) and f'(z) share the polynomial Q(1)(z)CM and if f(z) = Q2(z) whenever f'(z) = Q(2)(z), then f equivalent to f'. This result improves a result due to Li and Yi. Copyright (C) 2009 Jianming Qi et al.