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 $n$ $$p(z)=(z-z_m{)}^{t_m}(z-z_{m-1}{)}^{t_{m-1}}\cdots (z-z_0{)}^{t_0}(a_0+\sum _{\nu =\mu }^{n-(t_m+\cdots +t_0)}{a}_{\nu }{z}^{\nu })$$ where zeros $z_0,\dots ,z_m$ are in $\{z:|z|<1\}$ and the remaining $n-(t_m+\cdots +t_0)$ zeros are outside $\{z:|z|<k\}$ where $k\ge 1.$ Furthermore, we give a lower bound of this polynomial where zeros $z_0,\dots ,z_m$ are outside $\{z:|z|\le k\}$ and the remaining $n-(t_m+\cdots +t_0)$ zeros are in $\{z:|z|<k\}$ where $k\le 1.$

17 pages