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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


On bounded oscillation and asymptotic expansion of conformal strip mappings

Author: Arthur E. Obrock
Journal: Trans. Amer. Math. Soc. 173 (1972), 183-201
MSC: Primary 30A30
MathSciNet review: 0310214
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Abstract: Relations between the boundary parameters $ {\phi _ - },{\phi _ + }$ of a strip $ S = \{ {\phi _ - }(x) < y < {\phi _ + }(x)\} $ and the values $ f(x)$ of its canonical conformal mapping onto a horizontal strip $ H = \{ \vert\upsilon \vert < 1\} $ are studied. Bounded oscillation $ ({\max _y}\operatorname{Re} f(x + iy) - {\min _y}\operatorname{Re} f(x + iy) = \omega (x) = O(1))$ is characterized in terms of $ {\phi _ - },{\phi _ + }$. A formal series expansion $ \upsilon = \sum {y^m}{a_{m,n}}(x)$ is derived for the solution to the Dirichlet problem on $ S$ and its partial sums are used to obtain formulas for the asymptotic expansion of $ f$ in terms of $ {\phi _ + },{\phi _ - }$.

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PII: S 0002-9947(1972)0310214-4
Keywords: Conformal strip mappings, Ahlfors Distortion Theorem, Warschawski formula, the formula of Gol'dberg and Stročik, extremal length, Dirichlet problem, quasiconformal mapping, the Theorem of Teichmüller, Wittich and Belinski
Article copyright: © Copyright 1972 American Mathematical Society

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