On conformal rotations of simply-connected domains (Q1074735)

Let G be a bounded simply connected domain with \(0\in G\), and let f map \({\mathbb{D}}\) conformally onto G, with \(f(0)=0\). If \(g=f^{-1}\), the self map \(\phi_{\alpha}(z)=f(e^{i\alpha}g(z))\) of G onto G is called a conformal rotation of G. The reviewer had raised the question: If \(\phi_{\alpha_ n}(z)\Rightarrow z\), uniformly in G, for some null sequence \(\{\alpha_ n\}\), is it true that \(\phi_{\alpha}(z)\Rightarrow z\) for \(\alpha\) \(\to 0 ?\) The author constructs an intricate example to show that this is not always true.