The aim of this paper is the change of the conformal radius \(r(U)\) of a simply connected domain \(U \ni z_0\) versus the subdomain \(U_\epsilon\) which contains the points of distance \(\epsilon > 0 \) to \(\partial U\), where \(\epsilon\) is smaller than the distance from \(z_0\) to the boundary \(\partial U\). The main result is to show that the smallest exponent \(\lambda\) for which \[ r(U) - r(U_\epsilon) = 0(\epsilon^\lambda) \] satisfies \(0.59 < \lambda < 0.91 .\) There are also given relations to a conjecture of Brennan, Carleson and Jones and Kraetzer about integral means and the correct critical exponent. For more details about conformal radius and the mentioned conjecture see \textit{R. Kühnau} (ed.), Handbook of Complex Analysis: Geometric Function Theory, I (2002; Zbl 1057.30001), II (2005; Zbl 1056.30002).