Variation of the conformal radius

We study the change of the conformal radius r(U) of a simply connected planar domain U versus the subdomain U-epsilon consisting of the points of distance at least epsilon to partial derivativeU. We show that the smallest exponent lambda such that r(U) - r(U-epsilon) = O(epsilon(lambda)) satisfies 0.59 < lambda < 0.91. We also show that a well-known conjecture implies lambda= 2(root2 - 1).