The unique extremal QC mapping and uniqueness of Hahn-Banach extensions. (English) Zbl 1035.46501

Let \(M\) be the unit ball in the space \(L^\infty\) of essentially bounded complex valued measurable functions defined on the unit disc \(\Delta\) and for \(\varphi\in M\), let \(\Lambda_\varphi\) be the corresponding linear functional on the space \(L_a^1\) of analytic functions on \(\Delta\) in the space of \(L^1\)-integrable functions.
An outline of proof of the main steps of the proof of the following theorem is given: Let \(\varphi\in M\). If \(|\varphi|=k<1\) a.e. on \(\Delta\) and \(\|\Lambda_\varphi\|=\|\varphi\|_\infty\), then \(\varphi\) is the unique complex dilatation.


46E15 Banach spaces of continuous, differentiable or analytic functions
30C62 Quasiconformal mappings in the complex plane
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods