2022/07/06 by Huaying Wei, Katsuhiko Matsuzaki, Wei, Huaying +1
Mathematics · Medicine · #Analytic and geometric function theory #Anorectal Disease Treatments and Outcomes #Boundary (topology) #Chordal graph #Combinatorics #Complex Variables (math.CV) #Complex plane #FOS: Mathematics #Geometry #Graph #Infimum and supremum #Mathematical analysis #Mathematics #Plane (geometry) #Pure mathematics #Schwarzian derivative #math.CV
paper · pdf · doi:10.48550/arxiv.2207.02447
published in arXiv (Cornell University) (Cornell University)
arxiv created 2022/07/06 · openalex publication_date 2022/07/06 · arxiv updated 2022/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a univalent analytic function f on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on f, we show that there exists a chordal Loewner chain initiated from f until some finite time, and this Loewner chain defines a quasiconformal extension of f over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result developed on the disk by Becker and also the generalization of the Ahlfors-Weill formula. As an application of this quasiconformal extension, we complete the characterization of an element of the VMO-Teichmüller space on the half-plane using the vanishing Carleson measure condition induced by the (pre-)Schwarzian derivative.