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 ∈ D. For r > 0, let D_r be the connected component of D ∩ 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 ε-extension E_ε(a) of E among all continua E with given logarithmic capacity. Here, E_ε(a) = E ∪ z: |z-a| ≤ ε, α ∈ E, ε > 0.