On conformal mappings onto quasidisks
The paper under review studies the growth behavior of conformal mappings of the exterior \(\Delta^*=\mathbb{C}\cup\{\infty\}\setminus\overline{\Delta}\) of the unit disc \(\Delta\subset\mathbb{C}\). It is shown that if \(g:\Delta^*\to D\) is a conformal mapping with \(D\) a quasi-disc (and hence \(g\) has a \(K\)-quasiconformal extension to \(\mathbb{C}\cup\{\infty\}\)), then there are positive constants \(C=C(K)\), \(\beta=\beta(K)\) such that whenever \(z\in\Delta^*\cap\mathbb{C}\), \[ \text{dist}(g(z),\partial D)\leq (|z|^2-1)|g^\prime(z)| \leq C\, (|z|+1)^\beta \, \text{dist}(g(z),\partial D). \] A key tool used in the proof is the well-known fact that quasiconformal mappings of \(\mathbb{C}\cup\{\infty\}\) are quasisymmetric. As an application, the paper studies the near-boundary behavior of particular quasiconformal extensions of conformal mappings on bounded quasi-discs.
scientific article; zbMATH DE number 5723309
scientific article