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Conformal mappings and the distortion of moduli of rectangles


Author: Bruce Palka
Journal: Proc. Amer. Math. Soc. 50 (1975), 155-161
MSC: Primary 30A60
DOI: https://doi.org/10.1090/S0002-9939-1975-0369693-7
MathSciNet review: 0369693
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Abstract: In this paper it is shown that for an orientation-preserving homeomorphism $ f$ between domains $ D$ and $ D'$ in $ {\mathbf{\bar C}}$ to be conformal it is necessary and sufficient that there is an $ a > 1$ such that $ \bmod f(Q) = \bmod Q$ for each square $ Q$ in $ D$ and for each oriented rectangle $ Q$ in $ D$ with $ \bmod Q = a$.


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  • [1] C. A. Cazacu, Sur les inégalités de Rengel et la définition géométrique des représentations quasi-conformes, Rev. Romaine Math. Pures Appl. (9) 9 (1964), 141-155. MR.31 #4904. MR 0180673 (31:4904)
  • [2] F. W. Gehring, The definitions and exceptional sets for quasi-conformal mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 281 (1960), 28pp. MR 23 #A1800. MR 0124488 (23:A1800)
  • [3] F. W. Gehring and J. Väisälä, On the geometric definition for quasiconformal mappings, Comment. Math. Helv. 36 (1961), 19-32. MR 25 #4099. MR 0140684 (25:4099)
  • [4] J. A. Kelingos, Contributions to the theory of quasiconformal mappings, Thesis, University of Michigan, Ann Arbor, Mich., 1963.
  • [5] O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der math. Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 126, Springer-Verlag, Berlin and New York, 1965. MR 32 #5872. MR 0188434 (32:5872)
  • [6] O. Teichmüller, Untersuchungen über konforme und quasikonforme Abbildung, Deutsche Math. 3 (1938), 621-678.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0369693-7
Keywords: Conformal mapping, quasiconformal mapping, modulus of a quadrilateral
Article copyright: © Copyright 1975 American Mathematical Society

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