On the convergence of the Bieberbach polynomials in regions with corners

AbstractThe Bieberbach polynomials πn associated with a Jordan regionG and 0 ε G can be used to approximate the conformal mappingf0 fromG to{w: ¦w¦ < r0} withf0(0) = 0,f0′(0) = 1. Classical results of Keldych, Suetin, Mergelyan, and others give estimates of ∥f0-πn∥ under various assumptions on the smoothness of δG. More recently(*) $$\left\| {f_0 - \pi _n } \right\| = O(n^{ - \gamma } )$$ was proved for someγ>0 provided δG is piecewise smooth or a more general Jordan curve. One of our main results is that (*) holds for allγ < min(λ/(2−λ), 1/2) ifλπ (0<λ<2) is the smallest exterior angle in ∂G. Ifλ≤2/3 this upper bound forγ cannot be increased. Another theorem assumes thatf0 ε Lip α and the exterior mapψε Lipβ, 0<α, β≤1. The rate of convergence πn→ fo (n → ∞) is studied for a class of numerical test examples.