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

E. P. Dolzhenko

Sbornik. Mathematics, Tome 202 (2011) no. 12, pp. 1775-1823

Résumé

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.

Détail

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

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