Some inequalities for polynomials with a multiple zero at origin

If $P(z)$ is a polynomial of degree $n$ with origin as a zero of some multiplicity, then the estimate for $|P'(z)|$ in terms of $|P(z)|$ on $| z|=1$, when all other zeros of $P(z)$ lie in $|z|\le k,~ k\le 1$, is known [A. Aziz and W. M. Shah, \textit{Math. Inequal. Appl.,}~\textbf{7}(3)(2004),~379-391]. In this paper we prove some results in case $P(z)$ has a zero of multiplicity $s\ge 0$ at origin and all other zeros in $|z|\ge k, k\ge 1,$ as well as on $|z|=k , k\leq1$. We also consider s-fold lacunary polynomials of degree $n,$ by which we generalize some earlier well known results.