Minimization of the conformal radius under circular cutting of a domain

Let $D$ be a simply connected domain on the complex plane such that $0\in D$. For $r>0$, let $D_r$ be the connected component of $D\cap\{z:|z|<r\}$ containing the origin. For fixed $r$, we solve the problem on minimization of the conformal radius $R(D_r;0)$ among all domains $D$ with given conformal radius $R(D;0)$. This also leads to the solution of the problem on maximization of the logarithmic capacity of the local $\varepsilon$-extension $E_\varepsilon(a)$ of $E$ among all continua $E$ with given logarithmic capacity. Here, $E_\varepsilon(a)=E\cup{z:|z-a|\le\varepsilon}, a\in E,\varepsilon>0$.