Some Lp Inequalities for the Polar Derivative of a Polynomial

Abstract

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,∫2π0|Dαpneiθ|δdθ1/δ≤n|α|+1Fδ∫2π0|pneiθ|δdθ1/δ,where Fδ=(2π/∫2π0|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/z̄).