ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2019, Vol. 40, No. 9, pp. 1268–1274. c  Pleiades Publishing, Ltd., 2019. Asymptotics of Conformal Module of Nonsymmetric Doubly Connected Domain under Unbounded Stretching Along the Real Axis D. Dautova 1* and S. Nasyrov 1** (Submitted by F. G. Avkhadiev) 1 Kazan (Volga Region) Federal University, Kazan, Tatarstan, 420008 Russia Received May 11, 2019; revised May 14, 2019; accepted May 20, 2019 Abstract—We establish an asymptotic formula describing the behavior of the conformal module of a plane doubly connected domain under its stretching along the real axis with coefficient tending to infinity. The description is given in simple geometric terms, connected with equations of the boundary curves. Therefore, in the nonsymmetric case we give an answer to a problem by Prof. M. Vourinen. DOI: 10.1134/S1995080219090063 Keywords and phrases: conformal module, doubly-connected domain, convergence to a kernel. 1. INTRODUCTION One of the main characteristics of doubly-connected plane domain D is its conformal module m(D) (see, e.g. [2, 5, 9]). If the boundary components of D are non-degenerated, then (see, e.g., [4], ch. V, § 1) it can be mapped onto an annulus {r< |z| <R}. By definition, m(D) := 1/(2π) ln(R/r). The module can be calculated with the help of extremal lengths of curve-families (see, e.g. [1], Ch. 1). The module m(D) equals the extremal length of the family of curves in D connecting its boundary components; it is also is reciprocal to the extremal length of the family of curves in D separating its boundary components. Conformal modules of doubly connected domains are connected with conformal modules of quadri- laterals [1, 5]. A quadrilateral Q is a Jordan domain with four fixed points (vertices) on its boundary. The vertices separate the boundary into two pairs of opposite sides. Let us fix one of them. Then we will call the sides of the fixed pair marked. There exists a conformal mapping of Q onto a rectangle Π := [0,a] × [0,b] such that the marked sides are mapped onto the vertical sides of the rectangle Π. Then the conformal module of Q is m(Q) := a/b. The conformal module m(Q) coincides with the extremal length of the family of curves connecting in Q the marked sides. Conformal modules of doubly connected domains and quadrilaterals are conformally invariant. They are also quasiinvariant under quasiconformal mappings [1]. If f is an H -quasiconformal mapping, then 1 H m(G) ≤ m(f (G)) ≤ Hm(G), where G is either a doubly connected domain or a quadrilateral. Conformal modules are applied in investigation of various problems of the geometric function theory, in particular, in study of boundary properties of conformal and quasiconformal mappings, in investigation of quasiconformal mappings in the space, etc. Here we investigate the behavior of the conformal module of a doubly connected plane domain D with non-degenerated boundary components of a sufficiently arbitrary type under transform by the * E-mail: snasyrov@kpfu.ru ** E-mail: dautovadn@gmail.com 1268