On an upper bound for the polar derivative of a polynomial

Abstract

Liman, Mohopatra and Shah proved that if p(z) is a polynomial of degree n having no zeros in |z| < 1, then for all α,β ∈ C with |α| ≥ 1, |β| ≤ 1 and |z| = 1,

where Dαp(z) = np(z) + (α − z)p’(z) is the polar derivative of p(z) with respect to the point α. We extend and generalize this inequality for the polynomial p(z) which does not vanish in |z| < k, k ≤ 1. Our result also generalizes other known inequalities as well.