On the Maximum Modulus of Polynomials

A well-known result of 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 + 1)/2)n max|z| = 1 |p(z)|. In this paper, we prove a generalization and refinements of this result.