Some Inequalities for the Polar Derivative of Some Classes of Polynomials

TL;DR

This paper establishes new upper and lower bounds for the polar derivative of certain classes of polynomials with specified zero locations inside or outside the unit disk and outside or inside a circle of radius k.

Contribution

It introduces novel inequalities for the polar derivative of polynomials based on the zero distribution relative to the unit circle and a circle of radius k.

Findings

Derived an upper bound for the polar derivative with zeros inside the unit disk.

Established a lower bound for the polar derivative with zeros outside the circle of radius k.

Results extend existing inequalities to broader classes of polynomials with specified zero locations.

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 $k\leq 1.$