Approximation properties of the Bieberbach polynomials in closed Dini-smooth domains

Abstract

Let G be a finite Dini-smooth domain and w = phi(0)(z) be the confornial mapping of G onto D (0, r(0)) := {w :vertical bar w vertical bar < r(0)) with the normalization phi(0)(z(0)) = 0, phi'(0)(z(0)) = 1, where z(0) is an element of G. We investigate the approximation properties of the Bieberbach polynomials pi(n)(z) = 1,2,3(...) for the pair (G, z(0)) and estimate the error parallel to phi(0)-pi(n)parallel to((G) over bar) := max{vertical bar phi 0(z) - pi(n) (z) vertical bar: z is an element of (G) over bar} in accordance with the geometric parameters of (G) over bar.