Approximation properties of Julia polynomials

Abstract

Let G be a finite simply connected domain in the complex plane C, bounded by a rectifiable Jordan curve L, and let w = phi(0) (z) be the Riemann conformal mapping of G onto D(0, r (0)) := {w : vertical bar w vertical bar < r (0)}, normalized by the conditions phi(0) (z(0)) = 0, phi'(0) (z(0)) = 1. In this work, the rate of approximation of phi o (0) by the polynomials, defined with the help of the solutions of some extremal problem, in a closed domain (G) over bar is studied. This rate depends on the geometric properties of the boundary L.