The univalent Bloch-Landau constant, harmonic symmetry and conformal glueing

TL;DR

This paper improves the upper bound for the univalent Bloch-Landau constant by constructing special harmonically symmetric domains using conformal welding techniques.

Contribution

It introduces a new domain construction method based on harmonic symmetry and conformal welding to refine bounds on the Bloch-Landau constant.

Findings

Derived an improved upper bound for the univalent Bloch-Landau constant.

Established existence and properties of harmonically symmetric arcs in conformal domains.

Applied explicit solutions to the Polyá-Chebotarev problem in domain construction.

Abstract

By modifying a domain first suggested by Ruth Goodman in 1935 and by exploiting the explicit solution by Fedorov of the Poly\'a-Chebotarev problem in the case of four symmetrically placed points, an improved upper bound for the univalent Bloch-Landau constant is obtained. The domain that leads to this improved bound takes the form of a disk from which some arcs are removed in such a way that the resulting simply connected domain is harmonically symmetric in each arc with respect to the origin. The existence of domains of this type is established, using techniques from conformal welding, and some general properties of harmonically symmetric arcs in this setting are established.