Conformal images of tangential and nontangential arcs (Q1974413)

Let \(\varphi\) be a conformal map from the unit disk \(D\) into the complex plane. Let \(\gamma\) be a rectifiable Jordan arc in \(D\cup\{1\}\) that has a non-tangential approach in \(D\) to the point \(\zeta=1\). Let \(a_n={2^n-1\over 2^n+1}\), \(n=1,2,3,\dots\), let \(\gamma_n=\{z\in\gamma:a_n\leq|z|< a_{n+ 1}\}\), let \(|\gamma_n|\) denote the length of \(\gamma_n\), and let \(M_n= {|\gamma_n |\over a_{n+1}-a_n}\). The author proves that if the sequence \(\{M_n\}\) is bounded then either both \(\varphi(\gamma-\{1\})\) and \(\varphi ([0,1))\) are rectifiable or neither is rectifiable. The sharpness of this result is discussed. It is also proved that if \(\gamma\) is a Jordan arc in \(D\cup \{1\}\) that has a tangential approach in \(D\) to the point \(\zeta=1\), then there is a Jordan region \(\Omega\) and a conformal map \(\varphi\) from \(D\) onto \(\Omega\) such that \(\varphi([0,1]) =[0,1]\) and yet \(\varphi(\gamma)\) is not rectiviable.