Polynomial approximation of conformal maps

Let G be a finite domain, bounded by a Jordan curve Gamma, and let f(0) be a conformal map of G onto the unit disk. We are interested in the best rate of uniform convergence of polynomial approximation to f(0), in the case that Gamma is piecewise-analytic without cusps. In particular, we consider the problem of approximating f(0) by the Bieberbach polynomials pi(n) and derive results better than those in [5] and [6] for the case that the corners of Gamma have interior angles of the form pi/N. In the proof, the Lehman formulas for the asymptotic expansion of mapping functions near analytic corners are used. We study the question when these expansions contain logarithmic terms.