Browsing by Author "Osgood, B"

Now showing 1 - 5 of 5
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    Ellipses, near ellipses, and harmonic Mobius transformations
    (2005) Chuaqui, M; Duren, P; Osgood, B
    It is shown that an analytic function taking circles to ellipses must be a Mobius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic Mobius transformation.
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    Finding complete conformal metrics to extend conformal mappings
    (INDIANA UNIV MATH JOURNAL, 1998) Chuaqui, M; Osgood, B
    This paper shows how new differential geometric approaches to univalence criteria involving the Schwarzian derivative can be applied to a classical, but very general, criterion of Nehari. We show how positive solutions to the second order ODE associated to the Schwarzian can be used to construct complete conformal metrics. These lead to explicit formulas for homeomorphic and quasiconformal extensions of conformal mappings as generalizations of the Ahlfors-Weill extension.
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    General univalence criteria in the disk: Extensions and extremal function
    (SUOMALAINEN TIEDEAKATEMIA, 1998) Chuaqui, M; Osgood, B
    Many classical univalence criteria depending on the Schwarzian derivative are special cases of a result, proved in [18], involving both conformal mappings and conformal metrics. The classical theorems for analytic functions on the disk emerge by choosing appropriate conformal metrics and computing a generalized Schwarzian. The results in this paper address questions of extending functions which satisfy the general univalence criterion; continuous extensions to the closure of the disk, and homeomorphic and quasiconformal extensions to the sphere. The main tool is the convexity of an associated function along geodesics of the metric. The other important aspect of this study is an extremal function associated with a given criterion, along with its associated extremal geodesics. An extremal function for a criterion is one whose image is not a Jordan domain. An extremal geodesic joins points on the boundary which map to the same point in the image. We show that, for the general criterion, the image of an extremal geodesic under an extremal function is a euclidean circle.
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    John domains, quasidisks, and the Nehari class
    (WALTER DE GRUYTER & CO, 1996) Chuaqui, M; Osgood, B; Pommerenke, C
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    The Schwarzian derivative for harmonic mappings
    (2003) Chuaqui, M; Duren, P; Osgood, B