On a theorem of Haimo regarding concave mappings
Martin Chuaqui; Peter Duren; Brad Osgood
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2011)
- Volume: 65, Issue: 2
- ISSN: 0365-1029
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topMartin Chuaqui, Peter Duren, and Brad Osgood. "On a theorem of Haimo regarding concave mappings." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 65.2 (2011): null. <http://eudml.org/doc/289786>.
@article{MartinChuaqui2011,
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.},
author = {Martin Chuaqui, Peter Duren, Brad Osgood},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Concave mapping; Schwarzian derivative; Schwarzian norm; Haimo’s theorem; univalence; Sturm comparison; asymptotically conformal curve},
language = {eng},
number = {2},
pages = {null},
title = {On a theorem of Haimo regarding concave mappings},
url = {http://eudml.org/doc/289786},
volume = {65},
year = {2011},
}
TY - JOUR
AU - Martin Chuaqui
AU - Peter Duren
AU - Brad Osgood
TI - On a theorem of Haimo regarding concave mappings
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2011
VL - 65
IS - 2
SP - null
AB - 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.
LA - eng
KW - Concave mapping; Schwarzian derivative; Schwarzian norm; Haimo’s theorem; univalence; Sturm comparison; asymptotically conformal curve
UR - http://eudml.org/doc/289786
ER -
References
top- 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.
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