Quasiextremal distance domains and integrability of derivatives of conformal mappings (Q1337431)

Let \(f:\Omega \to \mathbb{D}\) be a conformal mapping from a simply connected domain \(\Omega\) into the open unit disk \(\mathbb{D}\). Let \(L\) be any straight line in the plane. The paper deals with the following problem: for which values of \(p\) is \[ \int_{L \cap \Omega} \bigl | f'(z) \bigr |^ pds < \infty? \tag{1} \] By an example due to Baernstein \(f'\) is not \(p\)-integrable for all \(1 \leq p<2\). The paper contains several geometric conditions, some on \(f(L)\) and some on \(\Omega\), which guarantee that (1) holds for all \(1 \leq p<2\). In particular it is proved: if \(\partial \Omega\) is a smooth (i.e. \(C^ 1\) class) Jordan curve or if \(\mathbb{D} \subset \Omega\) and \(f(\partial \mathbb{D})\) has both left and right tangents at each point and is of bounded rotation, then (1) holds for all \(1 \leq p<2\).