## Univalence criteria and quasiconformal extensions

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- by J. M. Anderson and A. Hinkkanen PDF
- Trans. Amer. Math. Soc.
**324**(1991), 823-842 Request permission

## Abstract:

Let $f$ be a locally univalent meromorphic function in the unit disk $\Delta$. Recently, Epstein obtained a differential geometric proof for the fact that if $f$ satisfies an inequality involving a suitable real-valued function $\sigma$, then $f$ is univalent in $\Delta$ and has a quasiconformal extension to the sphere. We give a more classical proof for this result by means of an explicit quasiconformal extension, and obtain generalizations of the result under suitable conditions even if $\sigma$ is allowed to be complex-valued and $\Delta$ is replaced by a quasidisk.## References

- Lars V. Ahlfors,
*Quasiconformal reflections*, Acta Math.**109**(1963), 291–301. MR**154978**, DOI 10.1007/BF02391816 - 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** - L. Ahlfors and G. Weill,
*A uniqueness theorem for Beltrami equations*, Proc. Amer. Math. Soc.**13**(1962), 975–978. MR**148896**, DOI 10.1090/S0002-9939-1962-0148896-1 - J. M. Anderson and A. Hinkkanen,
*A univalency criterion*, Michigan Math. J.**32**(1985), no. 1, 33–40. MR**777298**, DOI 10.1307/mmj/1029003129 - J. Becker,
*Conformal mappings with quasiconformal extensions*, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979) Academic Press, London-New York, 1980, pp. 37–77. MR**623464** - 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 - Olli Lehto,
*Univalent functions and Teichmüller spaces*, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR**867407**, DOI 10.1007/978-1-4613-8652-0 - Brad G. Osgood,
*Some properties of $f^{\prime \prime }/f^{\prime }$ and the Poincaré metric*, Indiana Univ. Math. J.**31**(1982), no. 4, 449–461. MR**662912**, DOI 10.1512/iumj.1982.31.31037 - Ch. Pommerenke,
*On the Epstein univalence criterion*, Results Math.**10**(1986), no. 1-2, 143–146. MR**869806**, DOI 10.1007/BF03322371

## Additional Information

- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**324**(1991), 823-842 - MSC: Primary 30C55; Secondary 30C62
- DOI: https://doi.org/10.1090/S0002-9947-1991-0994162-4
- MathSciNet review: 994162