A note on convex conformal mappings

Chuaqui, Martin; Osgood, Brad

doi:10.1090/proc/14429

A note on convex conformal mappings
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by Martin Chuaqui and Brad Osgood

Proc. Amer. Math. Soc. 147 (2019), 2655-2659

DOI: https://doi.org/10.1090/proc/14429

Published electronically: March 5, 2019

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Abstract:

We establish a new characterization for a conformal mapping of the unit disk $\mathbb {D}$ to be convex, and identify the mappings onto a half-plane or a parallel strip as extremals. We also show that, with these exceptions, the level sets of $\lambda$ of the Poincaré metric $\lambda |dw|$ of a convex domain are strictly convex.

References

Luis A. Caffarelli and Avner Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math. J. 52 (1985), no. 2, 431–456. MR 792181, DOI 10.1215/S0012-7094-85-05221-4
M. Chuaqui and B. Osgood, Ahlfors-Weill extensions of conformal mappings and critical points of the Poincaré metric, Comment. Math. Helv. 69 (1994), no. 4, 659–668. MR 1303231, DOI 10.1007/BF02564508
Martin Chuaqui, Peter Duren, and Brad Osgood, Schwarzian derivatives of convex mappings, Ann. Acad. Sci. Fenn. Math. 36 (2011), no. 2, 449–460. MR 2865506, DOI 10.5186/aasfm.2011.3628
Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
Seong-A. Kim and David Minda, The hyperbolic and quasihyperbolic metrics in convex regions, J. Anal. 1 (1993), 109–118. MR 1230512
Zeev Nehari, A property of convex conformal maps, J. Analyse Math. 30 (1976), 390–393. MR 440020, DOI 10.1007/BF02786725