Chordal Loewner families and univalent Cauchy transforms

Robert O. Bauer

    Research output: Contribution to journalArticlepeer-review

    Abstract

    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.

    Original languageEnglish (US)
    Pages (from-to)484-501
    Number of pages18
    JournalJournal of Mathematical Analysis and Applications
    Volume302
    Issue number2
    DOIs
    StatePublished - Feb 15 2005

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

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