A minimal area problem in conformal mapping II

Let S denote the usual class of functions f holomorphic and univalent in the unit disk U. For 0 < r < 1 and r(1 + r)^-2 < b < r(1 - r)^-2, let S(r, b) be the subclass of functions f ∈ S such that |f(r)| = b. In Theorem 1, we solve the problem of minimizing the Dirichlet integral in S(r, b). The first main ingredient of the solution is the establishment of sufficient regularity of the domains onto which U is mapped by extremal functions, and here techniques of symmetrization and polarization play an essential role. The second main ingredient is the identification of all Jordan domains satisfying a certain kind of functional equation (called "quadrature identities") which are encountered by applying variational techniques. These turn out to be conformal images of U by mappings of a special form involving a logarithmic function. In Theorem 2, this aspect of our work is generalized to encompass analogous minimal area problem when a larger number of initial data are prescribed.