Local Variations and Minimal Area Problem for Carathéodory Functions

Two kinds of minimal area problems have been studied. One, with analytic side conditions, was first treated by H. S. Shapiro. Another kind of problem initiated by A. W. Goodman deals with classes of analytic univalent functions with geometric constraints. The minimal area problem for the Carathéodory functions, i.e. analytic functions having positive real part in the unit disk, belongs to this second type. To solve it, we develop a technique in the frame of classical complex analysis that explores symmetrization type geometric transformations and local boundary variations. This method reduces the minimal area problem to a certain boundary value problem for analytic functions. In the case that the latter problem admits an explicit solution the original minimal area problem can be handled.