Geodesic
Criterion for the vanishing of the oscillation of the real part of a conformal mapping of strips
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1171-1195.

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Since 1976 it is known that the oscillation of the real part of a conformal mapping of strip domains asymptotically vanishes if and only if the respective extremal length is approximately additive. We show that these properties are equivalent to an explicit geometric condition of Ostrowski type introduced by Rodin and Warschawski in 1980. We also consider other equivalent conditions and deduce several known criteria from the main result.
Keywords: asymptotics, conformal mapping, isogonality condition of Ostrowski, $L$-strip, vanishing oscillation.
Mots-clés : strip domain 
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A. I. Parfenov. Criterion for the vanishing of the oscillation of the real part of a conformal mapping of strips. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1171-1195. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a140/

[1] J.B. Garnett, D.E. Marshall, Harmonic Measure, Cambridge Univ. Press, Cambridge, 2005 | MR | Zbl

[2] C. Gattegno, A. Ostrowski, “Représentation conforme à la frontière; domaines particuliers”, Mémorial Sci. Math., 110, Gauthier-Villars, Paris, 1949, 1–56 | MR | Zbl

[3] 1966 | MR | Zbl

[4] P. Henrici, Applied and Computational Complex Analysis, v. III, John Wiley Sons, New York, 1986 | MR | Zbl

[5] J.A. Jenkins, K. Oikawa, “Conformality and semi-conformality at the boundary”, J. Reine Angew. Math., 291 (1977), 92–117 | MR | Zbl

[6] R. Kühnau, “Bibliography of Geometric Function Theory”, Handbook of Complex Analysis: Geometric Function Theory, v. 2, ed. R. Kühnau, Elsevier, Amsterdam, 2005, 809–828 | DOI | MR | Zbl

[7] M. Lavrentiev, “On the continuity of univalent functions in closed domains”, C. R. (Doklady) Acad. Sci. U.S.S.R. New Ser., 4:5 (1936), 207–209 (Russian) | Zbl

[8] J. Lelong-Ferrand, Représentation conforme et transformations à intégrale de Dirichlet bornée, Gauthier-Villars, Paris, 1955 | MR | Zbl

[9] E. Lindelöf, “Sur la représentation conforme d'une aire simplement connexe sur l'aire d'un cercle”, Quatrième Congrès des Math. Scand. (1916), 1916, 59–90 | Zbl

[10] 1936 | Zbl

[11] A. Ostrowski, “Über den Habitus der konformen Abbildung am Rande des Abbildungsbereiches”, Acta Math., 64 (1935), 81–184 | DOI | MR | Zbl

[12] A. Ostrowski, “Zur Randverzerrung bei konformer Abbildung”, Prace Mat.-Fiz., 44 (1936), 371–471 | Zbl

[13] A. Ostrowski, Collected Mathematical Papers, v. 6, XIV: Conformal Mapping. XV: Numerical Analysis. XVI: Miscellany, Birkhäuser Verlag, Basel, 1985 | MR | Zbl

[14] A.I. Parfenov, “Error bound for a generalized M. A. Lavrentiev's formula via the norm of a fractional Sobolev space”, Siberian Electronic Mathematical Reports, 10 (2013), 335–377 (Russian) | MR | Zbl

[15] A.I. Parfenov, “Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem”, Siberian Advances in Mathematics, 27:4 (2017), 274–304 | DOI | MR | Zbl

[16] A.I. Parfenov, “Discrete Hölder estimates for a certain kind of parametrix. II”, Ufa Math. J., 9:2 (2017), 62–91 | DOI | MR

[17] A.I. Parfenov, “Approximate calculation of the defect of a Lipschitz cylindrical condenser”, Siberian Electronic Mathematical Reports, 15 (2018), 906–926 (Russian) | MR | Zbl

[18] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992 | MR | Zbl

[19] B. Rodin, S.E. Warschawski, “On conformal mapping of $L$-strips”, J. London Math. Soc. II. Ser., 11 (1975), 301–307 | DOI | MR | Zbl

[20] B. Rodin, S.E. Warschawski, “Extremal length and the boundary behavior of conformal mappings”, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 2 (1976), 467–500 | DOI | MR | Zbl

[21] B. Rodin, S.E. Warschawski, “Estimates for conformal maps of strip domains without boundary regularity”, Proc. London Math. Soc. III. Ser., 39 (1979), 356–384 | DOI | MR | Zbl

[22] B. Rodin, S.E. Warschawski, “On the derivative of the Riemann mapping function near a boundary point and the Visser-Ostrowski problem”, Math. Ann., 248:2 (1980), 125–137 | DOI | MR | Zbl

[23] J.L. Schiff, Normal Families, Springer-Verlag, New York, 1993 | MR | Zbl

[24] J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin, 1971 | MR | Zbl

[25] S.E. Warschawski, “On conformal mapping of infinite strips”, Trans. Amer. Math. Soc., 51 (1942), 280–335 | DOI | MR | Zbl

[26] S.E. Warschawski, “On the boundary behavior of conformal maps”, Nagoya Math. J., 30 (1967), 83–101 | DOI | MR | Zbl