SIMULTANEOUS APPROXIMATION OF THE RIEMANN CONFORMAL MAP AND ITS DERIVATIVES BY BIEBERBACH POLYNOMIALS

Let G be a domain in the complex plane C bounded by a rectiable Jordan curve Γ, let z0 ∈ G and let ϕ0 be the Riemann conformal map of G onto Dr := {w ∈ C : |w| < r}, normalized by ϕ0 (z0) = 0, ϕ 0 (z0) = 1.In this work the simultaneous approximations of ϕ0 and its derivatives by Bieberbach polynomials are investigated.The approximation rate in dependence of the smoothness parameters of the considered domains is estimated.