On a theorem of Haimo regarding concave mappings

Martin Chuaqui, Peter Duren, Brad Osgood

Abstract


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.


Keywords


Concave mapping; Schwarzian derivative; Schwarzian norm; Haimo’s theorem; univalence; Sturm comparison; asymptotically conformal curve

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References


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DOI: http://dx.doi.org/10.17951/a.2011.65.2.17-28
Data publikacji: 2016-07-27 21:54:07
Data złożenia artykułu: 2016-07-25 22:01:14

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