Some Lp Inequalities for the Polar Derivative of a Polynomial

Let pn z be a polynomial of degree n and Dα pn z its polar derivative. It has been proved that if pn z has no zeros in z < 1, then for δ ≥ 1 and α ≥ 1, 2π 0 Dα pn eiθ δdθ 1/δ ≤ n α + 1 Fδ 2π 0 pn eiθ δdθ 1/δ where Fδ = 2π/ 2π 0 1 + eiθ δdθ 1/δ. We also obtain analogous inequalities for the class of polynomials having all their zeros in z ≤ 1 and for the class of polynomials satisfying pn z ≡ znpn 1/ ¯z .