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

This article is cited in 2 scientific papers (total in 3 papers)

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

Full text: PDF file (848 kB)
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English version:
Sbornik: Mathematics, 2011, 202:12, 1775–1823

Bibliographic databases:

UDC: 517.54
MSC: 30C35, 30D04
Received: 03.11.2009 and 24.02.2011

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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