New Integral Mean Estimates for the Polar Derivative of a Polynomial

Abstract

Let p(z) be a polynomial of degree n and for α ∈ C, let D_αp(z):= np(z) + (α − z)p′(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain certain integral inequalities concerning the polar derivative of a polynomial, which besides yielding some interesting results, also includes some well-known theorems as special cases. Moreover we refine some Zygmund type inequalities for the polar derivative of a polynomial and present compact generalizations of some prior results.