INEQUALITIES FOR A POLYNOMIAL AND ITS DERIVATIVE

Let p(z) = ∑v = 0navzv be a polynomial of degree n having no zeros in |z| < k, k ≥ 1. Then we prove that for 1 ≤ r ≤ k, max|z| = r |p′(z)| ≤ n(r + k)n - 1 (1 + k)n max|z| = 1 |p(z)| and for 0 ≤ r ≤ λ ≤ 1, max|z| = r |p(z)| ≥ (k + r)n (k + λ)n max|z| = λ |p(z)|. Both results are best possible. Besides, it has been shown that if in addition p′(0) = 0, then the bound in the latter inequality can be considerably improved. © 1992.