Sets of values of systems of functionals in classes of univalent functions

The problem of describing the set of values of a system of functional $\{f(z),\dots,f^{(n)}(z)\}$ in the class of univalent functions holomorphic in the disk is formalized as a problem of constructing the set of attainability for a control system generated by the Löwner equation. In this problem the maximum principle turns out to be a necessary and sufficient condition for optimality. An algorithm for finding this set for a generalized Loewner equation with constant coefficients and continuous control is constructed. The results are extended to classes of bounded univalent functions.