Holomorphic mappings of the unit disk with finite angular derivative

We study the range of values of functions and its derivatives at an interior point of the unit disk under the condition of limited angular derivatives at fixed points on the boundary. An interval of values of the angular derivative is found, which corresponds to the existence of univalence domains in the unit disk for some classes of holomorphic mappings with boundary fixed points. These results are based on a description of the measures Alexandrov–Clark for the related classes of holomorphic mappings of the unit disk into itself.