Let *p(z)* be a polynomial of degree $n$, which has no zeros in *|z|< 1*, Dewan et al. [K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 (2010), pp. 38--41] established$$\Big|zp'(z)+\frac{n\beta}{2}p(z)\Big|\leq &\frac{n}{2}\Big\{\Big(\Big|\frac{\beta}{2}\Big|+\Big|1+\frac{\beta}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{\beta}{2}\Big|-\Big|\frac{\beta}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$for any $|\beta|\leq 1$ and *|z|=1*. In this paper we improve theabove inequality for the polynomial which has no zeros in $|z|< k, $ $ k\geq 1$, except $s$-fold zeros at the origin. Our resultsgeneralize certain well known polynomial inequalities.