The exact domain of univalence on the class of holomorphic maps of a disc into itself with an interior and a boundary fixed points

Abstract

We consider the problem of identifying domains of univalence on classes of holomorphic maps of the unit disc into itself. In 1926 E. Landau found the exact value of the radius of the disc of univalence on the class of such maps with a given value of the derivative at an interior fixed point. In 2017 V. V. Goryainov discovered the existence of univalence domains on classes of holomorphic maps of the unit disc into itself with an interior and a boundary fixed points, with a restriction on the value of the angular derivative at the boundary fixed point. However, the question of finding unimprovable domains of univalence remained open. In this paper, this extremal problem is solved completely: we find an exact univalence domain on the indicated class of holomorphic maps of the disc into itself. This result is a strengthening of Landau’s theorem for functions of the corresponding class.