On conformal moduli of polygonal quadrilaterals
The authors study the change of conformal moduli of polygonal quadrilaterals under certain geometric transformations. Results of this type have applications in the theory of quasiconformal mappings, see [\textit{O. Lehto} and \textit{K. I. Virtanen}, Quasiconformal mappings in the plane. Translated from the German by K. W. Lucas. 2nd ed. Die Grundlehren der mathematischen Wissenschaften. Band 126. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0267.30016)]. Examples of transformations considered in this paper are the motion of one vertex when others are kept fixed, rotation of sides, polarizations, symmetrizations, and averaging transformations. Some of the questions studied have been motivated by computational experiments, see [\textit{V. Heikkala, M. K. Vamanamurthy} and \textit{M. Vuorinen}, Comput. Methods Funct. Theory 9, No. 1, 75--109 (2009; Zbl 1198.33002)]. Inequalities concerning the behavior of the conformal modulus under these transformations are obtained. In addition, several examples and open problems are given.
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