A Note on a result due to Ankeny and Rivlin

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 $R\ge 1,$ $\displaystyle{\max_{|z|=R}|p(z)|}\leq \Big(\dfrac{R^n+1}{2}\Big)\displaystyle{\max_{|z|=1}|p(z)|}.$ In this paper, we consider polynomials with gaps, having all its zeros on the circle $S(0, K):={z: |z|=K}, ~0<K\le 1,$andestimatethevalueof$\Big(\dfrac{{\max_{|z|=R}|p(z)|}}{{\max_{|z|=1}|p(z)|}}\Big)^s$ for any positive integer $s\mathrm{.}$