Entire functions sharing a polynomial with their derivatives and normal families
Let ƒ be a nonconstant entire function and Q be a nonconstant polynomial. Let M be a positive number and k ≥ 2 be an integer. We prove that if ƒ and ƒ' share Q CM and |ƒ(k) (z)| ≤ M · (1 + |Q(z)|) whenever ƒ(z) = Q(z), then is constant. Furthermore, we prove the corresponding normality criterion.