Sets of Values of Systems of Functionals in Classes of Univalent Functions

Abstract

The problem of describing the set of values of a system of functional {f(Z), ..., 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 Loewner 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.