The geometric structure of regions, and direct theorems of the constructive theory of functions

Sufficient conditions and necessary conditions close to them are obtained for a bounded domain $G$ with Jordan boundary $L=\partial G$ to admit direct theorems of approximation theory in terms of the distance $\rho_{1+\frac1n}(z)$ from boundary points $z\in L$ to the $\bigl(1+\frac1n\bigr)$th level line of the function that maps the complement of the domain on the exterior of the unit disk.
Bibliography: 21 titles.