FEASIBILITY OF THE REICH PROCEDURE IN THE DECOMPOSITION OF PLANE QUASICONFORMAL MAPPINGS

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FEASIBILITY OF THE REICH PROCEDURE IN THE DECOMPOSITION OF PLANE QUASICONFORMAL MAPPINGS

In the decomposition problems, studied by Retch, of quasiconformal self mappings of the unit disc which keep the boundary points fixed, the construction of the first one requires the application of the Hahn-Banach theorem (so it is abstract) and it is only a variational decomposition (a small weight one), and that of the second one avoids the Hahn-Banach theorem and gets rid of the restriction to the variational decomposition. But the success of the second decomposition procedure (the Retch procedure) is guaranteed only when minimal maximal dilatation K(f) is sufficiently small. Therefore, it can not guarantee even a variational decomposition. Huang Xinzhong then proved that the inverse Reich procedure was successful for ally X(f). But the inverse Retch procedure is not so natural as the Retch procedure and the corresponding decomposition can not replace the first one. It is still an open problem whether the Reich procedure is successful for any X(f). The present paper gives an affirmative answer to this problem.