Plane domains with special cone condition

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.