A well-known result of Ankeney and Rivlin states that if p ( z ) is a polynomial of degree n , such that p ( z ) ≠ 0 in | z | < 1 , then max | z | = R ≥ 1 | p ( z ) | ≤ ( R n + 1 2 ) max | z | = 1 | p ( z ) | . In this paper we prove some generalizations and refinements of this result.