Quasidisks and twisting of the Riemann map

TL;DR

This paper investigates the integrability of derivatives of conformal maps from the unit disk onto quasidisks, identifying critical powers and relating findings to a circular analogue of Brennan's conjecture.

Contribution

It determines the range of critical complex powers for derivative integrability in quasidisks, advancing understanding of conformal map behavior.

Findings

Identifies critical powers for derivative integrability in quasidisks

Supports a circular analogue of Brennan's conjecture

Enhances understanding of conformal maps onto quasidisks

Abstract

Consider a conformal map from the unit disk onto a quasidisk. We determine a range of critical complex powers with respect to which the derivative is integrable. The results fit into the picture predicted by a circular analogue of Brennan's conjecture.