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 Mat. Sb., 2011, Volume 202, Number 12, Pages 57–106 (Mi msb7648)

Bounds for the moduli of continuity for conformal mappings of domains near their accessible boundary arcs

E. P. Dolzhenko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The paper presents bounds for the moduli of continuity $\omega(f,\overline{G},\delta)$ of conformal mappings $w=f(z)$ of a bounded simply connected domain $G$ with an arbitrary Jordan boundary onto a bounded simply connected domain with an arbitrary Jordan boundary, the ‘quality’ of boundaries being taken into account. For a Jordan curve (simple arc or a closed contour), its quality is characterized in general by its modulus of oscillation, and if it has finite length, by a more sensitive modulus of rectifiability — these purely metric concepts were introduced by the author in 1996. Theorems on the behaviour of conformal mappings of simply connected domains of arbitrary nature near open accessible boundary arcs are established.
Bibliography: 18 titles.

Keywords: univalent conformal mapping, accessible boundary arc of a simply connected domain, modulus of continuity, modulus of oscillation, modulus of rectifiability.

DOI: https://doi.org/10.4213/sm7648

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English version:
Sbornik: Mathematics, 2011, 202:12, 1775–1823

Bibliographic databases:

UDC: 517.54
MSC: 30C35, 30D04

Citation: E. P. Dolzhenko, “Bounds for the moduli of continuity for conformal mappings of domains near their accessible boundary arcs”, Mat. Sb., 202:12 (2011), 57–106; Sb. Math., 202:12 (2011), 1775–1823

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7648
• https://doi.org/10.4213/sm7648
• http://mi.mathnet.ru/eng/msb/v202/i12/p57

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This publication is cited in the following articles:
1. A. D. Baranov, K. Yu. Fedorovskiy, “Boundary regularity of Nevanlinna domains and univalent functions in model subspaces”, Sb. Math., 202:12 (2011), 1723–1740
2. M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Conditions for $C^m$-approximability of functions by solutions of elliptic equations”, Russian Math. Surveys, 67:6 (2012), 1023–1068
3. A. I. Aptekarev, P. A. Borodin, B. S. Kashin, Yu. V. Nesterenko, P. V. Paramonov, A. V. Pokrovskii, A. G. Sergeev, A. T. Fomenko, “Evgenii Prokof'evich Dolzhenko (on his 80th birthday)”, Russian Math. Surveys, 69:6 (2014), 1143–1148
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