Convergence of Bieberbach polynomials in domains with interior cusps

TL;DR

This paper investigates the convergence behavior of Bieberbach polynomials in domains with interior cusps, extending previous results and identifying conditions under which convergence or divergence occurs.

Contribution

It extends the understanding of Bieberbach polynomial convergence to domains with interior zero angles and constructs examples illustrating divergence at outward pointing cusps.

Findings

Bieberbach polynomials converge uniformly in certain domains with interior cusps.

A Keldysh-type example demonstrates divergence at outward pointing cusps.

The critical order of tangency determines convergence versus divergence.

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.