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

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

2008-06-13

arxiv.org/abs/0806.2282v1

Journal de Math\'ematiques Pures et Appliqu\'es Volume 92, Issue 4, October 2009, Pages 396-406

Mathematics

Complex Variables

Scientific paper

10.1016/j.matpur.2009.05.008

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.

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