Geodesic
Plane domains with special cone condition
Problemy analiza, Tome 3 (2014) no. 2, pp. 16-31.

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The paper considers the domains with cone condition in $\mathbb{C}$. We say that domain G satisfies the (weak) cone condition, if $p+V(e(p),H)\subset{G}$ for all $p\in{G}$, where $V(e(p),H)$ denotes right-angled circular cone with vertex at the origin, a fixed solution $\varepsilon$ and a height $H$, $0{H}\leq\infty$, and depending on the $p$ vector $e(p)$ axis direction. Domains satisfying cone condition play an important role in various branches of mathematic (e. g. [1], [2], [3] (p. 1076), [4]). In the paper of P. Liczberski and V. V. Starkov, $\alpha$–accessible domains were considered, $\alpha\in[0,1)$, — the domains, accessible at every boundary point by the cone with symmetry axis on $\{pt:t>1\}$. Unlike the paper of P. Liczberski and V. V. Starkov, here we consider domains, accessible outside by the cone, which symmetry axis inclined on fixed angle $\phi$ to the $\{pt: t>1\}$, $0\|\phi\|\pi/2$. In this paper we give criteria for this class of domains when the boundaries of domains are smooth, and also give a sufficient condition when boundary is arbitrary. This article is the full variant of [5], published without proofs.
Keywords: cone condition.
Mots-clés : $(\alpha,\beta)$–accessible domain 
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A. N. Anikiev. Plane domains with special cone condition. Problemy analiza, Tome 3 (2014) no. 2, pp. 16-31. http://geodesic.mathdoc.fr/item/PA_2014_3_2_a1/

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