Some Inequalities for the Polar Derivative of Some Classes of Polynomials

Abstract: In this paper, we investigate an upper bound of the polar derivative of a polynomial of degree $n$ $$p(z)=(z-z_m)^{t_m} (z-z_{m-1})^{t_{m-1}}\cdots (z-z_0)^{t_0}(a_0+\sum\limits_{\nu=\mu} ^{n-(t_m+\cdots+t_0)} a_{\nu}z^\nu)$$ where zeros $z_0,\ldots,z_m$ are in ${z:|z|<1}$ and the remaining $n-(t_m+\cdots+t_0 )$ zeros are outside ${z:|z|<k}$ where $k \geq 1.$ Furthermore, we give a lower bound of this polynomial where zeros $z_0,\ldots,z_m$ are outside ${z:|z|\leq k}$ and the remaining $n-(t_m+\cdots+t_0 )$ zeros are in ${z:|z|<k}$ where $k\leq 1.$