A continuous strictly increasing function μ mapping the real line onto itself is called ρ-quasisymmetric, 1 ≤ ρ < ∞, if 1/ρ ≤ μ(x + t) - μ (x)/μ(x)-μ(x-t)≤ ρ(1) for all x and all t ≠ 0. Beurling and Ahlfors first proved that any given ρ-quasisymmetrie funetion μ has an extension to a K-quasiconformal homeomorphismfrom the upper half-plane onto itself with K≤ρ~2.(2) Reed then improved the inequality (2) as follows: K < 8ρ.(3)In the present paper we give a detailed exposition of the computation for the inequality (2) (for [2], such an exposition may be concerned with by Reed because in [3] Lehto and Virtanen obtained K ≤ 8ρ(ρ + 1)~2 only) and prove the following result :
