Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1972-0310214-4
MathSciNet review: 0310214
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 _ - }$.


References [Enhancements On Off] (What's this?)

  • [1] L. V. Ahlfors, Untersuchungen zur Theorie der konformen Abbildungen und der ganzen Funktionen, Acta Soc. Sci. Fenn. Nova Series A 1 (1930), 1-40.
  • [2] -, Lectures on quasiconformal mappings, Van Nostrand Math. Studies, no. 10, Van Nostrand, Princeton, N. J., 1966. MR 34 #336. MR 0200442 (34:336)
  • [3] D. Drasin, On asymptotic curves of functions extremal for Denjoy's conjecture (to appear).
  • [4] B. G. Eke, Remarks on Ahlfors' distortion theorem, J. Analyse Math. 19 (1967), 97-134. MR 35 #6806. MR 0215971 (35:6806)
  • [5] D. Gaier, Estimates of conformal mappings near the boundary, Indiana Univ. Math. J. 21 (1972), 581-595. MR 0293072 (45:2151)
  • [6] A. A. Gol'dberg and T. V. Stročik, Conformal mapping of symmetric half-strips and angular regions, Litovsk, Mat. Sb. 6 (1966), 227-239. (Russian) MR 35 #5593. MR 0214744 (35:5593)
  • [7] J. A. Jenkins and K. Oikawa, On results of Ahlfors and Hayman, Illinois J. Math. 15 (1971), 664-671. MR 0296271 (45:5332)
  • [8] J. Lelong-Ferrand, Représentation conforme et transformations à intégrale de Dirichlet bornée, Gauthier-Villars, Paris, 1955. MR 16, 1096. MR 0069895 (16:1096b)
  • [9] O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der math. Wissenschaften in Einzeldarstellungen mit besonderer Berucksichtigung der Anwendungsgebiete, Band 126, Springer-Verlag, Berlin and New York, 1965. MR 32 #5872. MR 0188434 (32:5872)
  • [10] A. E. Obrock, Null Orlicz classes of Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A 498 (1972), 1-22. MR 0304648 (46:3780)
  • [11] E. Reich and H. Walczak, On the behavior of quasiconformal mappings at a point, Trans. Amer. Math. Soc. 117 (1965), 338-351. MR 31 #345. MR 0176070 (31:345)
  • [12] S. E. Warschawski, On conformal mapping of infinite strips, Trans. Amer. Math. Soc. 51 (1942), 280-335. MR 4, 9. MR 0006583 (4:9b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30A30

Retrieve articles in all journals with MSC: 30A30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0310214-4
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

American Mathematical Society