A Description Method in the Value Region Problem

Fedorov completely described the value region , , over the class of holomorphic univalent functions in the unit disk with real coefficients , . We extend this result to the class of bounded functions satisfying in . The problem is formulated as the reachable set problem for the Hamilton system of controllable differential equations in the frames of the Loewner theory. A family of Cauchy problems is substituted for the family of boundary value problems. The free parameter in the initial data serves as a parameter for the boundary curve of the value region. The algorithm proposed can be useful in other similar value region problems.