The author establishes the following result: Let \(G_1= e^{-h_1}\) be a quick admissible weight function and \(G_2= e^{-h_2}\) merely a fast weight function. Suppose that both weights lie in the range for which \((1-r^3)h_i'(r)\) remains bounded as \(r\to 1\), for each \(i= 1,2\). If \(h_1'(r)/h_2'(r)\to \infty\) as \(r\to 1\), then there is a self-map of the unit disk such that the induced composition operator \(C_\varphi\) maps \(A^2_{G_2}\) boundedly into itself but does not map \(A^2_{G_1}\) into itself. For fast weights \(G\), \(\varphi\in\zeta(G)\Rightarrow| \varphi'(\zeta)|\geq 1\) for all \(\zeta\in \partial\mathbb{D}\), where \(\mathbb{D}\) is the unit disk in the complex plane and \(\varphi'(\zeta)\) denotes the angular derivative of \(\varphi\) at \(\zeta\). As a consequence of the above result, there does not exist a fast weight for which there are no further restrictions.