In this paper, we study the growth of polynomials of degree $n$ having all their zeros on $|z|=k$, $k\leq 1$. Using the notation $M(p,t)=\max_{|z|=t}|p(z)|$, we measure the growth of $p$ by estimating $\big\{\frac{M(p,t)}{M(p,1)}\big\}^s$ from above for any $t\geq 1$, $s$ being an arbitrary positive integer. Also in this paper we improve the results recently proved by K. K. Dewan and Arty Ahuja [Growth of polynomials with prescribed zeros, J. Math. Ineq. 5 (2011), 355-361].