Boundary behavior of a conformal mapping (Q5896693)

Let \(D\) be a simply connected plane domain, not the whole plane. Let \(R^*\) denote those accessible boundary points of \(D\) such that \(D\) twists violently about them; that is, if \(\alpha\in R^*\) and \(w(\alpha)\) denotes its complex coordinate, then \[ \lim_{{w\to\alpha}\atop{w\in D}}\inf\arg(w-w(\alpha))=-\infty \hbox{ and }\lim_{{w\to\alpha}\atop{w\in D}}\sup\arg(w-w(\alpha))=+\infty, \] where \(\arg(w-w(\alpha))\) is defined and continuous in \(D\). We show that if a certain geometric condition holds at each point of a set \(W^*\subset R^*\), when \(W^*\) is a \(D\)-conformal null set. Let \(L_ \nu\) denote the ray with terminal point \(w(\alpha)\), \(\alpha\in R^*\), having inclination \(\nu\), \(0\leq\nu<2\pi\). Let m denote Lebesgue measure on \(L_ \nu\) and set \[ u(\nu)=\limsup_{r\to 0}{m((L_ \nu\cap D)\cap(w(\alpha),w(\alpha)+re^{i\nu})) \over r}. \] Let \(W^*=\{\alpha\in R^*:\) there exists \(L_{\nu_ i}\), \(i=1,2,3\), at \(w(\alpha)\) such that \(|\nu_ i-\nu_ j|=(2/3)\pi\), \(1\leq i<j\leq 3\), and \(u(\nu_ i)<1\) for \(i=1,2,3\}\). Theorem: \(W^*\) is a \(D\)-conformal null set.