INEQUALITIES FOR COMPLEX POLYNOMIAL WITH RESTRICTED ZEROS

Abstract

Let p(z) be a polynomial of degree n and for any complex number β, let D_{βp}(z)=np(z)+(β-z)p′(z) denote the polar derivative of the polynomial with respect to β. In this paper, we consider the class of polynomial

p(z)=(z-z₀)^{s} ❪ a₀+∑_{ν=0}^{n-s} a_{ν}z^{ν} ❫

of degree n having a zero of order s at z₀, |z₀|<1 and the remaining n-s zeros are outside |z|<k, k≥1 and establish upper bound estimates for the maximum of |D_{βp}(z)| as well as |p(Rz)-p(rz)|, R≥1≥1 on the unit disk.