Normal Families and Uniqueness of Entire Functions and Their Derivatives

Let f be a nonconstant entire function; let k . 2 be a positive integer; and let a be a nonzero complex number. If f(z) = a .. f'(z) = a, and f'(z) = a .. f (k)(z) = a, then either f = Ce .z + a or f = Ce .z + a(. - 1)/., where C and . are nonzero constants with . k-1 = 1. The proof is based on the Wiman.Valiron theory and the theory of normal families in an essential way.