Convergence of Bieberbach polynomials in domains with interior cusps

Abstract

We extend the results on the uniform convergence of Bieberbach polynomials to domains with certain interior zero angles (outward pointing cusps), and show that they play a special role in the problem. Namely, we construct a Keldysh-type example on the divergence of Bieberbach polynomials at an outward pointing cusp and discuss the critical order of tangency at this interior zero angle, separating the convergent behavior of Bieberbach polynomials from the divergent one for sufficiently thin cusps.