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- DATTORRO Mεβοο, Jon Dattorro, +93 authors Joe Zagami
- 2004

- BRAD OSGOOD, O. Lehto
- 2009

The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to conformal mappings of polygons bounded by circular arcs. More recently, Nehari [5, 6, 7] and others have developed important criteria for global univalence in terms of… (More)

- BRAD OSGOOD, DENNIS STOWE
- 2007

This note is a sequel to our paper [OS] where we generalized the Schwarzian derivative to conformal mappings of Riemannian manifolds. There we found that many of the phenomena familiar from the classical theory have counterparts in the more general setting. Here we advance this another step by giving a generalization of the well known univalence criterion… (More)

- M. Chuaqui, B. Osgood
- 1999

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… (More)

- B. G. OSGOOD, Zeev Nehari
- 2007

- Benjamin Van Roy, Hervé Lebret, +4 authors Brad Osgood

All Rights Reserved ii I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of… (More)

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. §1. Number of zeros. Consider the linear differential equation u ′′ (x) + p(x) u(x) = 0 , where p(x) = 1 (1 − x 2) 2 , (1) on the interval −1 < x < 1. Two… (More)

It is shown that an analytic function taking circles to ellipses must be a Möbius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic Möbius transformation. Analytic Möbius transformations take circles to circles. This is their most basic, most celebrated geometric property. We add the adjective 'analytic'… (More)

- By M. Chuaqui, P. Duren, B. Osgood, M. Chuaqui
- 2006

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass-Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformal metrics and on a… (More)