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

MathSciNet review:
1233965

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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.

**[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**, 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**, 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**, 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**, 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**, 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**, 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**, 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**, 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**, 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**, 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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DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1233965-7

Article copyright:
© Copyright 1995
American Mathematical Society