Sharp domains of univalence for some classes of holomorphic self-mappings of the disc

The problem of finding domains of univalence on classes of holomorphic self-mappings of the disc is concidered. In 1926, E. Landau found sharp radius of the disc of univalence on the class of such mappings with a given value of derivative at the inner fixed point. In 2017, V. Goryainov discovered the existence of domains of univalence on classes of holomorphic self-mappings of the disc with two fixed points and conditions on the values of angular derivatives at the boundary fixed points. The report is devoted to the development of these results. An asymptotically sharp two-sided estimate of the domains of univalence on classes of holomorphic self-mappings of the disc that keep the real diameter and have a restriction on the value of the product of angular derivatives at the boundary fixed points is obtained in collaboration with O. Kudryavtseva. Sharp domain of univalence on the class of holomorphic self-mappings of the disc with inner and boundary fixed points and a restriction on the value of angular derivative at the boundary fixed point is found. The last result improves Landau theorem for functions from the corresponding class.