A well known result due to Ankeny and Rivlin [1] states that if $p(z)={sum}^n_{j=0}a_jz^j$ is a polynomial of degree n satisfying $p(z){ eq}0$ for |z| < 1, then for $R{geq}1$ $$max_{{mid}z{mid}=R}{mid}p(z){mid}{leq}{frac{R^n+1}{2}}max_{{mid}z{mid}=1}{mid}p(z){mid}$$. It was proposed by Professor R.P. Boas Jr. to obtain an inequality analogous to this inequality for polynomials having no zeros in |z| < k, k > 0. In this paper, we obtain some results in this direction, by considering polynomials of degree $n{geq}2$, having all its zeros on the disk |z| = k, $k{leq}1$.