On the polar derivative of a polynomial

For a polynomial $p(z)=a_nz^n+\sum_{\nu=\mu}^na_{n-\nu}z^{n-\nu},\‎ ‎1\leq \mu\leq n$ of degree $n$‎, ‎having all zeros in $|z|\leq k,\‎ ‎k\leq 1$‎, ‎Dewan et al [K‎. ‎K‎. ‎Dewan‎, ‎N‎. ‎Singh and A‎. ‎Mir‎, ‎Extension‎ ‎of some polynomial inequalities to the polar derivative‎, ‎J‎. ‎Math‎. ‎Anal‎. ‎Appl‎. ‎352 (2009) 807-815] proved that‎ $$ \max_{|z|=1}|D_{\alpha}p(z)|\geq‎ ‎\frac{n}{1+k^{\mu}}\{ (|\alpha|-A_{\mu})‎ ‎\max_{|z|=1}|p(z)|+‎‎ \frac{|\alpha|k^{\mu}+A_{\mu}}{k^n}‎ ‎\min_{|z|=k}|p(z)|\}‎, $$ ‎‎where $|\alpha|\geq k^{\mu}$ and $ A_{\mu}= \frac{n(\mid‎ ‎a_{n}\mid-\frac{m}{k^{n}}) k^{2 \mu}+\mu \mid a_{n‎- ‎\mu}\mid k^{\mu‎ -‎1}}{n(\mid a_{n}\mid-\frac{m}{k^{n}})\mid k^{\mu‎ -‎1}+\mu \mid a_{n‎- ‎\mu}\mid}.$ In this paper we improve and extend the above‎ ‎inequality‎. ‎Our result generalizes certain well-known polynomial‎ ‎inequalities‎.