On Polynomials and Their Polar Derivative

Abstract

Let P (z) be a polynomial of degree n and for any complex number α, let D α P (z) = nP (z) + (α − z)P (z) denote the polar derivative of P (z) with respect to α. Here, we consider the class of polynomials P (z) = a n z n + n ν=µ a n−ν z n−ν , 1 ≤ µ ≤ n, having all zeros in |z| ≤ k, k ≤ 1 and thereby establish several interesting results regarding the integral mean estimates for the polar derivative of P (z). Our results not only generalize and refine some known polynomial inequalities, but also a variety of interesting results can be deduced from these by a fairly uniform procedure.