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

Full-text PDF

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]**L. V. Ahlfors,*Sufficient conditions for quasiconformal extensions*, Discontinuous Groups and Riemann Surfaces (Leon Greenberg, ed.), Ann. of Math. Stud., no. 79, Princeton Univ. Press, Princeton, NJ, 1974. MR**0374415 (51:10615)****[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), 823-842. MR**994162 (91h:30026)****[B-S]**S. Bergman and M. Schiffer,*Kernel functions and conformal mappings*, Compositio Math.**8**(1951), 205-249. MR**0039812 (12:602c)****[Ca]**K. Carne,*Schwarzian derivative for conformal maps*, J. Reine Angew. Math.**408**(1990), 10-33. MR**1058982 (91h:30040)****[Ch1]**M. Chuaqui,*The Schwarzian derivative and quasiconformal reflections on*, Ann. Acad. Sci. Fenn. Ser. A I Math.**17**(1992), 315-326. MR**1190327 (94a:53036)****[Ch2]**-,*On a theorem of Nehari and quasidiscs*, Ann. Acad. Sci. Fenn. Ser. A I Math.**18**(1993), 117-124. MR**1207899 (94f:30026)****[Ch3]**-,*Ricci curvature and a criterion for simple-connectivity on the sphere*, Proc. Amer. Math. Soc. (to appear). MR**1197534 (95a:53059)****[Ch-O1]**M. Chuaqui and B. Osgood,*The Schwarzian derivative and conformally natural quasiconformal extensions from one to two to three dimensions*, Math. Ann.**292**(1992), 267-280. MR**1149035 (94d:30031)****[Ch-O2]**-,*Sharp distortion theorems associated with the Schwarzian derivative*, J. London Math. Soc. (2)**48**(1993), 289-298. MR**1231716 (94g:30005)****[Ep]**C. Epstein,*The hyperbolic Gauss map and quasiconformal reflection*, J. Reine Angew. Math.**372**(1986), 96-135. MR**863521 (88b:30029)****[Ne1]**Z. Nehari,*The Schwarzian derivative and schlicht functions*, Bull. Amer. Math. Soc.**55**(1949), 545-551. MR**0029999 (10:696e)****[Ne2]**-,*Univalence criteria depending on the Schwarzian derivative*Illinois J. of Math.**23**(1979), 345-351. MR**537795 (80i:30033)****[O-S1]**B. Osgood and D. Stowe,*The Schwarzian derivative and conformal mappings of Riemannian manifolds*, Duke Math. J.**67**(1992), 57-99. MR**1174603 (93j:53062)****[O-S2]**-,*A generalization of Nehari's univalence criterion*, Comment. Math. Helv.**65**(1990), 234-242. MR**1057241 (92a:53015)****[Sa]**M. Sakai*The sub-mean value property of subharmonic functions and its applications to estimate the Gaussian curvature of the span metric*, preprint.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
30C35,
30C62

Retrieve articles in all journals with MSC: 30C35, 30C62

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1233965-7

Article copyright:
© Copyright 1995
American Mathematical Society