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Asymptotics of the Exterior Conformal Modulus of a Quadrilateral under Stretching Map

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Abstract

In this paper, we focus on studying the distortion of the exterior conformal modulus of a quadriilateral of a sufficiently arbitrary form under the stretching map along the abscissa axis with coefficient \(H \to \infty \). 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 \(({\text{1/}}\pi )\ln H\) as \(H \to \infty \). Therefore, we give a solution to the Vuorinen problem for the exterior conformal modulus of a sufficiently arbitrary quadrilateral.

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Funding

The work of the first author is supported by the Development Program of the Scientific and Educational Mathematical Center of the Volga Federal District (agreement no. 075-02-2022-882).

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Authors and Affiliations

  1. Kazan Federal University, 420008, Kazan, Russia

    S. R. Nasyrov & G. V. Nguyen

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  1. S. R. Nasyrov
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  2. G. V. Nguyen
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Correspondence to S. R. Nasyrov or G. V. Nguyen.

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Nasyrov, S.R., Nguyen, G.V. Asymptotics of the Exterior Conformal Modulus of a Quadrilateral under Stretching Map. Russ Math. 67, 66–71 (2023). https://doi.org/10.3103/S1066369X23050080

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  • DOI: https://doi.org/10.3103/S1066369X23050080

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