On the maximum modulus of a polynomial and its derivatives

Abstract

Let f(z) be an arbitrary entire function and M(f,r)=max|z|=r|f(z)|. For a polynomial P(z), having no zeros in |z|<k, k≥1, Bidkham and Dewan (1992) proved max|z|=r|P′(z)|≤(n(r+k)n−1/(1+k)n)max|z|=1|P(z)| for 1≤r≤k. In this paper, we generalize as well as improve upon the above inequality.