Feasibility of the Reich procedure in the decomposition of plane quasiconformal mappings (Q1842122)

Let \(Q\) be the class of quasiconformal mappings of the unit disc \(U = \{| z | < 1\}\) onto itself, and \(Q_ I\) be the subclass of \(Q\), whose elements keep the boundary points of \(U\) fixed. \textit{E. Reich} [Comment. Math. Helv. 53, 15-27 (1978; Zbl 0369.30017)] studied the problem of decomposing \(f \in Q_ I\) into \(f = f_ 2 \circ f_ 1\) with \(f_ i \in Q_ I\), and \(K(f_ i) < K(f)\), \(i = 1,2\). He proves that the decompositon of \(f \in Q_ I\) can always be done by a procedure using the Hahn-Banach theorem. In the paper, he proves by another procedure without the use of the Hahn-Banach theorem that when \(f \in Q_ I\), and \[ K(f) < M = {3 + \sqrt{5}\over 2} \tag{*} \] there exist \(f_ i \in Q_ I\), \((i = 1,2)\), such that \(f = f_ 2 \circ f_ 1\), and \[ K(f_ i) < K(f)^{3/2} - K(f) + 1.\tag{**} \] In this paper, the author proves that the condition \((*)\) is not necessary. In other words, this procedure is feasible for any \(f\in Q_ I\) wiht \(1 \leq K(f) < +\infty\). But the bound for \(K(f_ i)\) corresponding to \((**)\) will be more complicated.