On a theorem of Haimo regarding concave mappings

Martin Chuaqui; Peter Duren; Brad Osgood

Displaying similar documents to “On a theorem of Haimo regarding concave mappings”

On a theorem of Haimo regarding concave mappings

Martin Chuaqui, Peter Duren, Brad Osgood (2011)

Annales UMCS, Mathematica

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A relatively simple proof is given for Haimo's theorem that a meromorphic function with suitably controlled Schwarzian derivative is a concave mapping. More easily verified conditions are found to imply Haimo's criterion, which is now shown to be sharp. It is proved that Haimo's functions map the unit disk onto the outside of an asymptotically conformal Jordan curve, thus ruling out the presence of corners.

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The standard conformal compactification of Euclidean space is the round sphere. We use conformal geodesics to give an elementary proof that this is the only possible conformal compactification.