Inequalities for a Polynomial and Its Derivative II

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 consider the polynomial p(z) = a0 + Σnv = μaυzυ having all its zeros in |z| ≤ k > 1 and obtain a generalization of this result. Our result improves upon a result recently proved by Bidkham and Dewan (J. Math. Anal. Appl.166 (1992), 19-324).