A note on \(\mu(Z)\)-homeomorhpisms (Q5943060)

Since the work of \textit{A. Beurling} and \textit{L. V. Ahlfors} [Acta Math. 96, 125-142 (1956; Zbl 0072.29602)] the interplay of quasisymmetric functions on \(R\) and quasiconformal mappings \(f\) of the upper half space \(H\) onto itself has been well-known. For a homeomorphism \(f\) of \(H\), locally quasiconformal in \(H\), the author imposes conditions on the dilatation of \(f\) to obtain bounds for the quasisymmetric function \(h(x,t)= \max(h, h^{-1})\) of \(f\) on \(R\) where \(x\in R\), \(t> 0\) and \(h= [f(x+ t)- f(x)]/[f(x)- f(x- t)]\). These extend author's earlier results in [\textit{J. Chen}, \textit{Z. Chen} and \textit{C. He}, Mich. Math. J. 43, No. 2, 211-220 (1996; Zbl 0863.30029)].