A unified approach to univalence criteria in the unit disc
Author:
Martin Chuaqui
Journal:
Proc. Amer. Math. Soc. 123 (1995), 441-453
MSC:
Primary 30C35; Secondary 30C62
DOI:
https://doi.org/10.1090/S0002-9939-1995-1233965-7
MathSciNet review:
1233965
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Abstract | References | Similar Articles | Additional Information
Abstract: From the recent injectivity criterion of Osgood and Stowe we recover many of the known univalence criteria in the unit disc D and derive as well new conditions on D and simply-connected domains. While the criteria of Epstein can be established in this fashion, we show how the 'diameter term' in the criterion of Osgood and Stowe gives a sharper form of a condition of Ahlfors. Finally, on simply-connected domains we find a sufficient condition for univalence that is the counterpart to a necessary one proved by Bergman and Schiffer.
- [Ah1] Lars V. Ahlfors, Sufficient conditions for quasiconformal extension, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Princeton Univ. Press, Princeton, N.J., 1974, pp. 23–29. Ann. of Math. Studies, No. 79. MR 0374415
- [Ah2]
-, Schwarzian derivative and cross-ratio in
, Complex Analysis: A Collection of Papers Dedicated to Albert Pluger, Birkhäuser, Boston, MA, 1989.
- [A-H] J. M. Anderson and A. Hinkkanen, Univalence criteria and quasiconformal extensions, Trans. Amer. Math. Soc. 324 (1991), no. 2, 823–842. MR 994162, https://doi.org/10.1090/S0002-9947-1991-0994162-4
- [B-S] S. Bergman and M. Schiffer, Kernel functions and conformal mapping, Compositio Math. 8 (1951), 205–249. MR 0039812
- [Ca] Keith Carne, The Schwarzian derivative for conformal maps, J. Reine Angew. Math. 408 (1990), 10–33. MR 1058982, https://doi.org/10.1515/crll.1990.408.10
- [Ch1] Martin Chuaqui, The Schwarzian derivative and quasiconformal reflections on 𝑆ⁿ, Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), no. 2, 315–326. MR 1190327, https://doi.org/10.5186/aasfm.1992.1720
- [Ch2] Martin Chuaqui, On a theorem of Nehari and quasidiscs, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 1, 117–124. MR 1207899
- [Ch3] Martin Chuaqui, Ricci curvature and a criterion for simple-connectivity on the sphere, Proc. Amer. Math. Soc. 122 (1994), no. 2, 479–485. MR 1197534, https://doi.org/10.1090/S0002-9939-1994-1197534-9
- [Ch-O1] Martin Chuaqui and Brad Osgood, The Schwarzian derivative and conformally natural quasiconformal extensions from one to two to three dimensions, Math. Ann. 292 (1992), no. 2, 267–280. MR 1149035, https://doi.org/10.1007/BF01444621
- [Ch-O2] M. Chuaqui and B. Osgood, Sharp distortion theorems associated with the Schwarzian derivative, J. London Math. Soc. (2) 48 (1993), no. 2, 289–298. MR 1231716, https://doi.org/10.1112/jlms/s2-48.2.289
- [Ep] Charles L. Epstein, The hyperbolic Gauss map and quasiconformal reflections, J. Reine Angew. Math. 372 (1986), 96–135. MR 863521, https://doi.org/10.1515/crll.1986.372.96
- [Ne1] Zeev Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551. MR 0029999, https://doi.org/10.1090/S0002-9904-1949-09241-8
- [Ne2] Zeev Nehari, Univalence criteria depending on the Schwarzian derivative, Illinois J. Math. 23 (1979), no. 3, 345–351. MR 537795
- [O-S1] Brad Osgood and Dennis Stowe, The Schwarzian derivative and conformal mapping of Riemannian manifolds, Duke Math. J. 67 (1992), no. 1, 57–99. MR 1174603, https://doi.org/10.1215/S0012-7094-92-06704-4
- [O-S2] Brad Osgood and Dennis Stowe, A generalization of Nehari’s univalence criterion, Comment. Math. Helv. 65 (1990), no. 2, 234–242. MR 1057241, https://doi.org/10.1007/BF02566604
- [Sa] M. Sakai The sub-mean value property of subharmonic functions and its applications to estimate the Gaussian curvature of the span metric, preprint.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1995-1233965-7
Article copyright:
© Copyright 1995
American Mathematical Society