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Abstract. We establish a relationship between Strebel boundary dilatation of a quasisymmetric function of the unit circle and indicated by the change in the module of the quadrilaterals with vertices on the circle. By using general theory of universal Teichmüller space, we show that there are many quasisymmetric functions of the circle have the property that the smallest dilatation for a quasiconformal extension of a quasisymmetric function of the unit circle is larger than indicated by the change in the module of quadrilaterals with vertices on the circle. Mathematics Subject Classification (1991). Primary 32G15; Secondary 30C60; 30C75.Keywords. Boundary dilatation, quasisymmetric function, quasiconformal mapping, Strebel point. §1. IntroductionIn this paper, the following notation will be used. C = the finite complex plane; ∆ = {z ∈ C; |z| < 1}; Γ = ∂∆ (boundary of ∆);∆ = ∆∪Γ; ∆ r = {z; r < |z| < 1}, where 0 < r < 1; H = the upper half plane; R = the real line in C.

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