Browsing by Author "Duren, Peter"

Now showing 1 - 5 of 5
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    OSCILLATION OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS
    (CAMBRIDGE UNIV PRESS, 2009) Chuaqui, Martin; Duren, Peter; Osgood, Brad; Stowe, Dennis
    In this note we study the zeros of solutions of differential equations of the form u '' + pu = 0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.
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    Schwarzian derivative criteria for valence of analytic and harmonic mappings
    (2007) Chuaqui, Martin; Duren, Peter; Osgood, Brad
    For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass-Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally, certain classes of harmonic mappings are shown to have finite Schwarzian norm.
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    SCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS
    (SUOMALAINEN TIEDEAKATEMIA, 2011) Chuaqui, Martin; Duren, Peter; Osgood, Brad
    A simple proof is given for Nehari's theorem that an analytic function f which maps the unit disk onto a convex region has Schwarzian norm parallel to f parallel to <= 2. The inequality in sharper form leads to the conclusion that no convex mapping with parallel to f parallel to = 2 can map onto a quasidisk. In particular, every bounded convex mapping has Schwarzian norm parallel to f parallel to < 2. The analysis involves a structural formula for the pre-Schwarzian of a convex mapping, which is studied in further detail.
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    SCHWARZIAN NORMS AND TWO-POINT DISTORTION
    (2011) Chuaqui, Martin; Duren, Peter; Ma, William; Mejia, Diego; Minda, David; Osgood, Brad
    An analytic function f with Schwarzian norm parallel to gf parallel to <= 2(1 + delta(2)) is shown to satisfy a pair of two-point distortion conditions, one giving a lower bound and the other an upper bound for the deviation. Conversely, each of these conditions is found to imply that parallel to gf parallel to <= 2(1 + delta(2)). Analogues of the lower bound are also developed for curves in R-n and for canonical lifts of harmonic mappings to minimal surfaces.
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    TWO-POINT DISTORTION THEOREMS FOR HARMONIC MAPPINGS
    (2009) Chuaqui, Martin; Duren, Peter; Osgood, Brad
    In earlier work, the authors have extended Nehari's well-known Schwarzian derivative criterion for univalence of analytic functions to a univalence criterion for canonical lifts of harmonic mappings to minimal surfaces. The present paper develops some quantitative versions of that result in the form of two-point distortion theorems. Along the way some distortion theorems for curves in R(n) are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form.