We investigate asymptotical behavior of the conformal module of a doubly-connected domain which is the difference of two homothetic rectangles under stretching it along the abscissa axis. Thereby, we give the answer to a question put by Prof. M. Vuorinen. 2010 AMS Subject Classification. Primary 30C20.

We study the point pair function in subdomains G of $${\mathbb {R}}^n$$ R n . We prove that, for every domain $$G\subsetneq {\mathbb {R}}^n$$ G ⊊ R n , this function is a quasi-metric with the constant less than or equal to $$\sqrt{5}/2$$ 5 / 2 . Moreover, we show that it is a metric in the domain $$G={\mathbb {R}}^n{\setminus }\{0\}$$ G = R n \ { 0 } with $$n\ge 1$$ n ≥ 1 . We also consider generalized versions of the point pair function, depending on a parameter $$\alpha >0$$ α > 0 , and show that in some domains these generalizations are metrics if and only if $$\alpha \le 12$$ α ≤ 12 .

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We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in terms of the Lauricella hypergeometric function. This result enables us to understand the dissimilarities of the exterior and interior of a convex polygonal quadrilateral. We also give a Mathematica algorithm for the computation. In particular, we study the special case of an isosceles trapezoidal polygon L and obtain some estimates for the coefficient of quasiconformal reflection over L in terms of special functions and geometric parameters of L.