The boundary of a simply connected domain at an inner tangent point (Q811494)

Let \(T^*\) be the set of accessible boundary points at which the inner tangent to \(\partial D\) exists. That is, if \(a\in T^*\) and \(w(a)\) represents its complex coordinate, then there exists a unique \(\nu(a)\), \(0\leq\nu(a)<2\pi\), such that for each \(\varepsilon>0\), \((\varepsilon<\pi/2)\), there exists a \(\delta>0\) such that \[ \Delta=\{w(a)+\rho e^{i\nu};\;0<\rho<\delta,\;|\nu- \nu(a)|<{\pi\over 2}-\varepsilon\}\subset D \hbox{ and } w\to a \hbox{ as } w\to w(a),\;w\in\Delta. \] Let \(\gamma(a,r)\) represent the unique component of \(D\cap\{| w-w(a)|=r\}\) that intersects the inner normal \(\{w(a)+\rho e^{i\nu(a)}: \rho>0\}\), \(L(a,r)\) denote the length of \(\gamma(a,r)\) and set \(A(a,r)=\int_ 0^ r L(a,r')dr'\). Finally let \(AD^*\) be those points of \(T^*\) at which a non-zero angular derivative exists. Our main result is a purely geometrical proof of a theorem that describes the boundary of \(D\) near \(a\in T^*\). As a consequence we have (1) a geometric description of the boundary of \(D\) near almost every \(a\in AD^*\) that is a generalization of the geometric behavior of a smooth curve; (2) an answer to \(T^*\) and hence on \(AD^*\) of the two open questions and conjectures made by McMillan concerning the length and area ratios \({L(a,r)/2\pi r}\) and \(A(a,r)/ \pi r^ 2\) as \(r\to 0\).