Quadrilaterals and extremal quasiconformal extensions

We show that the smallest maximal dilatation for a quasiconformal extension of a quasisymmetric function of the unit circle may be larger than indicated by the change in the module of the quadrilaterals with vertices on the circle. The research of the second author has been partially supported by the Alfred P. Sloan Foundation and by the U.S. National Science Foundation grant DMS 94-00999. This research was completed while the first author was visiting the University of Illinois at Urbana-Champaign. He wishes to thank the Department of Mathematics for its kind hospitality.