Asymptotics of the exterior conformal modulus of an arbitrary quadrilateral under stretching map

Abstract

In this paper, we focus on studying the distortion of the exterior conformal modulus of a quadrilateral of sufficiently arbitrary form under the stretching map along the abscissa axis with coefficient H → ∞. By using the properties of quasiconformal transformations and taking into account some facts from the theory of elliptic integrals, we confirm that the asymptotic behavior of this modulus does not depend on the shape of the boundary of the quadrilateral. Especially, it is equivalent to (1/π) log H as H → ∞. Therefore, we give a solution to the Vuorinen problem for the exterior region of an arbitrary quadrilateral.