A unified approach to univalence criteria in the unit disc
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- by Martin Chuaqui
- Proc. Amer. Math. Soc. 123 (1995), 441-453
- DOI: https://doi.org/10.1090/S0002-9939-1995-1233965-7
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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.References
- Lars V. Ahlfors, Sufficient conditions for quasiconformal extension, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 23–29. MR 0374415 —, Schwarzian derivative and cross-ratio in ${R^n}$, Complex Analysis: A Collection of Papers Dedicated to Albert Pluger, Birkhäuser, Boston, MA, 1989.
- J. M. Anderson and A. Hinkkanen, Univalence criteria and quasiconformal extensions, Trans. Amer. Math. Soc. 324 (1991), no. 2, 823–842. MR 994162, DOI 10.1090/S0002-9947-1991-0994162-4
- S. Bergman and M. Schiffer, Kernel functions and conformal mapping, Compositio Math. 8 (1951), 205–249. MR 39812
- Keith Carne, The Schwarzian derivative for conformal maps, J. Reine Angew. Math. 408 (1990), 10–33. MR 1058982, DOI 10.1515/crll.1990.408.10
- Martin Chuaqui, The Schwarzian derivative and quasiconformal reflections on $S^n$, Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), no. 2, 315–326. MR 1190327, DOI 10.5186/aasfm.1992.1720
- 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
- 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, DOI 10.1090/S0002-9939-1994-1197534-9
- 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, DOI 10.1007/BF01444621
- 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, DOI 10.1112/jlms/s2-48.2.289
- Charles L. Epstein, The hyperbolic Gauss map and quasiconformal reflections, J. Reine Angew. Math. 372 (1986), 96–135. MR 863521, DOI 10.1515/crll.1986.372.96
- Zeev Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551. MR 29999, DOI 10.1090/S0002-9904-1949-09241-8
- Zeev Nehari, Univalence criteria depending on the Schwarzian derivative, Illinois J. Math. 23 (1979), no. 3, 345–351. MR 537795
- 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, DOI 10.1215/S0012-7094-92-06704-4
- Brad Osgood and Dennis Stowe, A generalization of Nehari’s univalence criterion, Comment. Math. Helv. 65 (1990), no. 2, 234–242. MR 1057241, DOI 10.1007/BF02566604 M. Sakai The sub-mean value property of subharmonic functions and its applications to estimate the Gaussian curvature of the span metric, preprint.
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- 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