Integral Mean Estimates for a Polynomial with Restricted Zeros

Abstract

Let P n be the class of polynomials P (z) = n j =0 a j z j of degree n. For P ∈ P n , defineandIt was shown by Turan [14], that if P ∈ P n and P (z) has all its zeros in |z| ≤ 1, thenThe result is best possible and equality holds for P (z) = αz n +β, where |α| = |β|. As a generalization of inequality (11.1), Malik [12] proved that if P ∈ P n and P (z) = 0 in |z| ≤ k, where k ≤ 1 thenThe result is best possible and equality holds for P (z) = (z + k) n .A. Liman ( )