Univalence criteria and quasiconformal extensions

Anderson, J. M.; Hinkkanen, A.

doi:10.1090/S0002-9947-1991-0994162-4

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.

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