Covering of radial segments for domains bounded by \(k\)-circles (Q1187088)

Consider ring domains \(D\) which have inner boundary \(\{| w|=1\}\) and outer boundary a closed curve \(\gamma\) on the sphere which is a \(k\)- circle (quasiconformal) in the sense of D. Blevins. Let \(\theta\) be fixed and consider the line of slope \(\theta\) through the origin which intersects \(\gamma\) at \(w_ 1=r_ 1e^{i\theta}\), \(w_ 2=r_ 2e^{i(\theta + \pi)}\). The author mimimizes the functionals \(r_ 1 r_ 2\) and \(1/r_ 1 + 1/r_ 2\). By using elementary quasiconformal theory and symmetrization, the author shows that the minima occur (up to rotation) when \(D\) is the image of a wedge region under a specific Möbius transformation. In particular, \(\gamma\) must pass through infinity. \{Remark: The author states his problem in terms of conformal mappings from a normalized annulus onto \(D\), but the problem seems to be purely geometric\}.