On a theorem of Haimo regarding concave mappings

Martin Chuaqui, Peter Duren, Brad Osgood


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.


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

Full Text:



Ahlfors, L. V., Weill, G., A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975-978.

Becker, J., Pommerenke, Ch., Uber die quasikonforme Fortsetzung schlichter Funktionen, Math. Z. 161 (1978), 69-80.

Chuaqui, M., Duren, P. and Osgood, B., Schwarzian derivatives of convex mappings, Ann. Acad. Sci. Fenn. Math. 36 (2011), 449-460.

Chuaqui, M., Duren, P. and Osgood, B., Schwarzian derivative criteria for univalence of analytic and harmonic mappings, Math. Proc. Cambridge Philos. Soc. 143 (2007), 473-486.

Chuaqui, M., Duren, P. and Osgood, B., Concave conformal mappings and prevertices of Schwarz-Christoffel mappings, Proc. Amer. Math. Soc., to appear.

Chuaqui, M., Osgood, B., Sharp distortion theorems associated with the Schwarzian derivative, J. London Math. Soc. 48 (1993), 289-298.

Duren, P. L., Univalent Functions, Springer-Verlag, New York, 1983.

Duren, P., Schuster A., Bergman Spaces, American Mathematical Society, Providence, Rhode Island, 2004.

Gabriel, R. F., The Schwarzian derivative and convex functions, Proc. Amer. Math. Soc. 6 (1955), 58-66.

Haimo, D. T., A note on convex mappings, Proc. Amer. Math. Soc. 7 (1956), 423-428.

Nehari, Z., The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551.

Nehari, Z., Some criteria of univalence, Proc. Amer. Math. Soc. 5 (1954), 700-704.

Nehari, Z., A property of convex conformal maps, J. Analyse Math. 30 (1976), 390-393.

Pommerenke, Ch., On univalent functions, Bloch functions and VMOA, Math. Ann. 236 (1978), 199-208.

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


  • There are currently no refbacks.

Copyright (c) 2016 Martin Chuaqui