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An extremal problem for quasiconformal mappings in an annulus


Author: Alvin M. White
Journal: Proc. Amer. Math. Soc. 71 (1978), 267-274
MSC: Primary 30A38; Secondary 30A60
DOI: https://doi.org/10.1090/S0002-9939-1978-0480981-9
MathSciNet review: 0480981
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Abstract: The following extremal problem is solved. We consider a family of continuously differentiable univalent quasiconformal mappings $w = f(z)$ of the annulus $r < |z| < 1$ onto the unit disk minus some continuum containing the origin. For a point b on a fixed circle, maximize $|f(b)|$ within the family. The problem is solved by using a variational method due to Schiffer. The extremal function and the maximum are found in terms of the Weierstrass $\wp$-function and the elliptic modular function.


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Article copyright: © Copyright 1978 American Mathematical Society