Uniformization by rectangular domains: A path from slits to squares

Let Σ(Ω) be the class of functions [Formula presented] univalent on a finitely connected domain Ω, ∞∈Ω⊂C‾. By a classical result due to H. Grötzsch, the function f₀ maximizing ℜa₁ over the class Σ(Ω) maps Ω onto C‾ slit along horizontal segments. Recently, M. Bonk found a similar extremal problem, which maximizer f₁∈Σ(Ω) maps Ω onto a domain on C‾, whose complementary components are squares. In this note, we discuss a parametric family of extremal problems on the class Σ(Ω) with maximizers f_m, 0<m<1, mapping Ω onto domains on C‾, whose complementary components are rectangles with horizontal and vertical sides and with module m.