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l. lntroduction. The standard construction for a quasiconformal extension of a real quasisymmetric function å is the one introduced by Beurling and Ahlfors [l]. They proved that a q-quasisymmettic h has a g2-quasiconformal extension to the upper half-plane H. A careful examination of Beurling's and Ahlfors's estimation shows that pz can be replaced by Q't' for g close to one and by 3pzl4 fot large a I4l. T. Reed [6] has given the bound 8g, which is better than the previous ones for large g. The bound 2q has been announced by wan-cai Lai [3], but the cor-

The dilatation of Beurling-Ahlfors extensions of quasisymmetric functions