Asymptotics of the Conformal Modulus of Unbounded Symmetric Doubly Connected Domain Under Stretching

Abstract

Abstract: We describe the asymptotic behavior of the conformal modulus of an unbounded doubly connected planar domain, symmetric with respect to the coordinate axes, when stretched in the direction of the abscissa axis with coefficient tending to infinity. Therefore, we give a partial answer to a problem, suggested by M. Vourinen, for the case of unbounded domains. We also use classical theorems by Ahlfors and Warshavskii to investigate the asymptotical behavior of the conformal modulus of a quadrilateral under stretching.