The inclusion of classical families in the closure of the universal Teichmüller space (Q1199264)

The author studies the closure of the universal Teichmüller space. He works with the following model of this space: \({\mathcal T}=\{\phi=(f''/f')| f\) conformal on the unit disc \(\mathbb{D}\) with quasiconformal extension to the Riemann sphere \(\hat\mathbb{C}\}\). Let \({\mathcal L}=\{f| f\) conformal on \(\mathbb{D}\), \(f(0)=0\) and \(f'(0)=1\) and \(\mathbb{C}- f(\mathbb{D})\) is the union of closed half lines such that the corresponding open half lines are disjoint.\} If \(f\in{\mathcal L}\) then there exists an \(h\) conformal on \(\mathbb{D}\) such that \(h(\mathbb{D})\) is a convex region and \(\text{Re}\{f''/f'\}>0\). Let \({\mathcal L}_ 0=\{f\in{\mathcal L}|\) the corresponding \(h(\mathbb{D})\) has a boundary with constant turning tangent.\} The main result of the paper is: \(\forall f\in{\mathcal L}_ 0: f''/f'\in\text{cl}(\mathcal T)\). Using the fact that \(\mathcal T\) can be continuously embedded in the Bers model \(\mathcal J\) of the universal Teichmüller space it is proved that for all \(f\in{\mathcal L}_ 0\) the Schwarzian derivative \(\{f,z\}\) is an element of \(\text{cl}(\mathcal J)\).