An improvement and a generalization of Ankeny and Rivlin’s result on the maximum modulus of polynomials

Abstract

For an arbitrary entire function f(z), let M(f, r) = max|z|=r|f(z)|. By considering the polynomial of degree n having no zero in the interior of the unit circle|z| = 1, Ankeny and Rivlin obtained M(p, R) ≤ Rn + 1/2 M(p, 1), R ≥ 1. In this paper, we consider the polynomial of degree n having no zero in |z| < k, k ≥ 1 and simultaneously considering the sᵗʰ derivative, 0 ≤ s < n, of the polynomial, we have obtained an improvement as well as a generalization of Ankeny and Rivlin’s result.