Extremal area problems for conformal mappings and their applications (Q1909843)

The authors consider some external area problems and estimates related to areas of domains obtained by conformal mappings of certain canonical domains, such as disk, an annulus, and a disk with concentric slits. They proved some new theorems concerning the area minimization at the conformal mapping of certain domains and obtained the new proof of a well-known Carleman's inequality [\textit{T. Carleman}, Über ein Minimalproblem der mathematischen Physik, Math. Z. 1, 208--212 (1918; JFM 46.0765.02)]. Also they managed to apply their results on a conformal mapping to establish the uniqueness of solution of the exterior inverse boundary value problem as stated by \textit{F. D. Gakhov} [Boundary value problems. 3rd ed. Moskva: Izdat. Nauka (1977; Zbl 0449.30030), Chapter IV]. Some of the proofs are instructive and ingenious but on the whole the article is a rather technical one.