A Note on a result due to Ankeny and Rivlin

Abstract: Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$ having no zeros in the unit disk. ~Then it is well known that for $R\geq 1,$ $\displaystyle{\max_{|z|=R}|p(z)|}\leq \Big(\dfrac{R^n+1}{2}\Big)\displaystyle{\max_{|z|=1}|p(z)|}.$ In this paper, we consider polynomials with gaps, having all its zeros on the circle $S(0, K):={z: |z|=K}, ~0<K\le 1,$~ and estimate the value of $\Big(\dfrac{{\max_{|z|=R}|p(z)|}}{{\max_{|z|=1}|p(z)|}}\Big)^s$ for any positive integer $s.$