Almost quasiconformal mappings with given boundary values and a complex dilatation bound (Q1112202)

E. Reich first studied the extremal problem of quasiconformal mappings with given boundary correspondence and dilatation bound, and obtained the necessary and sufficient conditions for a quasiconformal mapping to be extremal. The dilatation bound function b(w) is required to satisfy \(0<b_ 0\leq b(w)\leq b_ 1<1.\) Later the reviewer [Chin. Ann. Math., Ser. A 7, 465-473 (1986; Zbl 0641.30023)] discussed the problem of relaxing the condition to \(0\leq b(w)\leq 1\), but under some additional conditions of continuity to b(w). In this paper the author proves that the additional conditions can be given up. The main result of this paper is as follows: Let F be an almost quasiconformal self mapping of the unit disc U, \(T(\bar T\subset U)\) be a measurable set and b(w) (0\(\leq b(w)\leq 1)\) be a measurable function on T, which can have countably many singularity points and whose increasement near the singularity points cannot be too quick. Denote by A the family of almost all quasiconformal mappings which satisfy: (i) \(G(e^{i\theta})=F(e^{i\theta})\), (ii) \(| G_{\bar w}/G_ w| \leq b(w)\), \(w\in T.\) Set \(k_ G= \sup_{w\in U\setminus T}| G_{\bar w}/G_ w|\). If \(G_ 0\in A\) and \(k_{G_ 0}=\inf_{G\in A}k_ G\), we say that \(G_ 0\) is extremal within A. Set \(T_ 0=\{w\in T|\) \(b(w)=0\}\) and \[ \tau (z)=\kappa (z)/b(G^{-1}(z)),\quad z\in U\setminus G(T_ 0),\quad \tau(z)=0,\quad z\in G(T_ 0), \] where \(\kappa(z)=G_{\bar z}^{-1}/G_ z^{-1}.\) Theorem. The necessary and sufficient condition for G to be extremal within A is: Either \(k_ G=0\), or \(k_ G>0\) and \[ \sup_{\phi \in B,\| \phi \|_{U\setminus G(T_ 0)}=1}| \iint_{U}\tau (z)\phi (z)dx dy| =1, \] where B is the Banach space of all analytic L' functions in U.