On the polar derivative of lacunary type polynomials

Let $p(z)=a_nz^n+\sum_{l=\nu}^na_{n-l}z^{n-l}$, where $1\leq \nu \leq n$, be a polynomial of degree $n$ having all its zeros in $|z|\leq k\leq 1$. For polar derivative $D_{\alpha}p(z)$, it is known that for each $|\alpha|\leq 1$ on $|z|=1$, \begin{align*} |D_{\alpha}p(z)|\leq \frac{n}{1+k^{\nu}}\Big\{(|\alpha|+k^{\nu})\|p\|_{\infty}-\frac{1-|\alpha|}{k^{n-\nu}}\min_{|z|=k}|p(z)|\Big\}. \end{align*} In this paper, we obtain the $L_q$ mean extension and a refinement of the above and other related results for the polar derivative of polynomials.