The polar derivative of a polynomial $P(z)$ of degree $n$ with respect to a complex number $\gamma$ is a polynomial $nP(z)+(\gamma-z)P'(z)$ of degree at most $n-1$ and is denoted by $D_{\gamma}P(z)$. We consider the class of polynomials $P(z)=a_0+\sum_{v=\mu}^na_v z^v$, $\mu\geq 1,$ of degree $n$ such that $P(z)\neq 0$ in $|z|<k$, $k\geq 1$ and establish some upper bound estimates for the maximum modulus of $D_\gamma P(z)$ on the unit disk by involving some of the coefficients of $P(z)$. The obtained results refine and generalize some well known polynomial inequalities.