The Behavior of Conformal Maps of Domains Near Convex Boundary Arcs

The behavior of the derivatives of conformal maps of the unit disk onto simply connected domains in the complex plane whose boundaries contain convex or concave attainable arcs, as well as the behavior of the derivatives of the inverse maps, is studied. It is proved that these derivatives exist and are bounded on the corresponding arcs and near them; a criterion for their continuity at points of these arcs is stated and proved.