A REAL.ANALYTIC QUASICONFORMAL EXTE|{SIO| { OF A QUASISYMMETRIC

for all rcal x and t, t+0. It is well-known that every p-quasisymmetric function can be extended to a K-quasiconformal mapping f: H * H, where I / denotes the upper half-plane. This was first shown by Beurling and Ahlfors [2] who gave an explicit construction for an extension I This extension is in the class C1(I1). In the present note we shall introduce an analytic kernel into the integerals which define / and thus obtain a real-analytic solution of the boundary value problem. The construction follows closely the one in [2]. Special attention, however, is paid to the estimation of the maximal dilatation K. The elementary but laborious com-putations in [2] leading to the estimate K<qz fot q>l seem to contain some slips. In fact, the same computations yield a somewhat sharper bound. Remark. Recently Lehto [4] has demonstrated the existence of a real-analytic solution of the boundary value problem for quasiconformal mappings by a different, less explicit method. The maximal dilatation of his solution seems to be essentially larger than that of ours. We state our result as Theorem. Euery p-quasisymmetric function has a K-quasiconformal real-analytic extension to the upper half-plane. There is a number Qo(:1.925 057...) such that K<.qslz if l=p-.qo and K<3qzl4 if q=-qo. Proof. Let q>l arrd rp be a fixed q-quasisymmetric function. (Since a l-quasisymmetric function is linear, there is nothing to prove in the case Q:1.) Our construction is based on the functions uy: H*R defined b