L^p inequalities for polar derivatives of polynomials

Abstract

Let p(z) = ∑n v=0 avz v be a polynomial of degree n and for α ∈ C , let Dαp(z) = np(z) + (α − z)p′(z) denote the polar derivative of the polynomial p(z) with respect to α . It is well known that the polar derivative generalizes the ordinary derivative. In this paper, we obtain Lp inequalities for polar derivatives of polynomials satisfying p(z) ≡ znp( z ) and for polynomials satisfying p(z) ≡ znp( z ) . Our results generalize several results in this direction. Mathematics subject classification (1991): Primary 26D05, 30D15. Secondary 41A17.