Inequalities for the polar derivative of a polynomial

Some Lp Inequalities for the Polar Derivative of a Polynomial

“…The polynomial D α p n has been called by Laguerre [6] the "émanant" of p n , by Pólya and Szegö [9] the "derivative of p n with respect to the point α," and by Marden [8] simply "the polar derivative of p n ." It is obviously of interest to obtain estimates concerning growth of D α p n z and one such estimate is due to Aziz [1], who extended the inequality (2) due to Lax [7] for D α p n by proving Theorem A. If p n z is a polynomial of degree n having no zeros in the disk z < 1 then for every real or complex number α with α ≥ 1 we have…”

A comparison inequality for rational functions

“…Despite the similarity in the statements, the proofs used in the polynomial case (see [1,12]) could not be directly applied in the rational case. We will explain more on the differences in the proofs in the next section.…”

On an inequality concerning the polar derivative of a polynomial

“…If we let p → ∞ in (2), we get inequality (1). Let D α P(z) denote the polar differentiation of polynomial P(z) with respect to a real or complex number α. Then…”

“…Aziz [1] extended (13) to the polar derivative of a polynomial and proved that if P(z) is a polynomial of degree n which does not vanish in |z| < 1, then for every complex number α with |α| ≥ 1,…”