Sharp univalent covering domain for the class of holomorphic self-maps of a disc with fixed interior and boundary points

The class of holomorphic maps of the unit disc to itself, with an interior and a boundary fixed point is under consideration. For the class of such functions a sharp univalent covering domain is found in its dependence on the value of the angular derivative at the boundary fixed point and the position of the interior fixed point. This result can be viewed as a refinement of Landau's theorem on the univalent covering disc for the class of bounded holomorphic functions with prescribed derivative at the interior fixed point.