A MINIMAL AREA PROBLEM IN CONFORMAL MAPPING II By DOV AHARONOV, HAROLD S. SHAPIRO AND ALEXANDER YU. SOLYNIN* Abstract. Let S denote the usual class of functions .r 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 ] E S such that I f(r)l = 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 ingredi- ent is the identification ofatl Jordan domains satisfying a certain kind of functional equation (called "quadrature identities") which are encountered by applying vari- ational 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. 1 Introduction 1.1 Let U be the open unit disk of the complex plane. For f holomorphic in U, the Dirichlet integral of f is D(f) =jo If'(re~O)l~rdrdO I/'12 &r, where da denotes planar measure. Since oo (1,1) O(f) = ~r ~-'~ nla.I 2 _> 7r n'=l for (1.2) f(z) = z + a2z 2 +... , *The third author thanks for its hospitality the Mittag-Leffier Institute of Royal Swedish Academy of Sciences where this work was finalized. This author was supported in part by the Swedish Institute and by the Russian Fund for Fundamental Research, grant no. 97-01-00259. 259 JOURNAL D'ANALYSE MATHI~MATIQUE, Vol. 83 (2001)