A Note on a result due to Ankeny and Rivlin

Abstract

Let $p(z)=a_0+a_{1}z+a_2{z}^2+a_3{z}^3+\cdots +a_n{z}^n$ be a polynomial of degree $n$ having no zeros in the unit disk. ~Then it is well known that for $R\ge 1,$ $\underset{|z|=R}{\max }|p(z)|\le \left(\frac{R^n+1}{2}\right)\underset{|z|=1}{\max }|p(z)|.$ In this paper, we consider polynomials with gaps, having all its zeros on the circle $S(0,K):=\{z:|z|=K\},0<K\le 1,$~ and estimate the value of $(\frac{{\max }_{|z|=R}|p(z)|}{{\max }_{|z|=1}|p(z)|}{)}^s$ for any positive integer $s.$