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.

我们考虑将单位圆盘的全纯映射类上的单价域识别为自身的问题。1926 年，E. Landau 在内部不动点的导数给定值的情况下，在此类地图上找到了单价圆盘半径的精确值。2017 年，VV Goryainov 发现在单位圆盘的全纯映射类上存在单价域，它具有内部和边界不动点，对边界不动点处的角导数值有限制。然而，寻找不可改进的单价域的问题仍然存在。在本文中，这个极值问题得到了彻底的解决：我们在圆盘的全纯映射的指示类上找到了一个精确的单价域。