2003/06/07 by Robert O. Bauer, Bauer, Robert O. · 1 citation
Computer Science · Mathematics · #30C45 #30E20 #60E10 #Bayesian Methods and Mixture Models #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Probability (math.PR) #Random Matrices and Applications #math.CV #math.PR #msc:30C45 #msc:30E20 #msc:60E10
paper · pdf · doi:10.48550/arxiv.math/0306130
29 pages
arxiv created 2003/06/07 · openalex publication_date 2003/06/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study chordal Loewner families in the upper half-plane and show that they have a parametric representation. We show one, that to every chordal Loewner family there corresponds a unique measurable family of probability measures on the real line, and two, that to every measurable family of probability measures on the real line there corresponds a unique chordal Loewner family. In both cases the correspondence is being given by solving the chordal Loewner equation. We use this to show that any probability measure on the real line with finite variance and mean zero has univalent Cauchy transform if and only if it belongs to some chordal Loewner family. If the probability measure has compact support we give two further necessary and sufficient conditions for the univalence of the Cauchy transform, the first in terms of the transfinite diameter of the complement of the image domain of the reciprocal Cauchy transform, and the second in terms of moment inequalities corresponding to the Grunsky inequalities.