Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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.


References [Enhancements On Off] (What's this?)

  • [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 $ {R^n}$, 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 $ {S^n}$, 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.

Similar Articles

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

American Mathematical Society