Uniformization by rectangular domains: A path from slits to squares

Let $\Sigma(\Omega)$ be the class of functions<br>$f(z)=z+\frac{a_1}{z}+\cdots$ univalent on a finitely connected<br>domain $\Omega$, $\infty\in \Omega\subset \overline{\mathbb{C}}$.<br>By a classical result due to H.~Gr\"{o}tzsch, the function $f_0$<br>maximizing $\Re \,a_1$ over the class $\Sigma(\Omega)$ maps<br>$\Omega$ onto $\overline{\mathbb{C}}$ slit along horizontal<br>segments. Recently, M.~Bonk found a similar extremal problem,<br>which maximizer $f_1\in \Sigma(\Omega)$ maps $\Omega$ onto a<br>domain on $\overline{\mathbb{C}}$, whose complementary components<br>are squares. In this note, we discuss a parametric family of<br>extremal problems on the class $\Sigma(\Omega)$ with maximizers<br>$f_m$, $0<m<1$, mapping $\Omega$ onto domains on<br>$\overline{\mathbb{C}}$, whose complementary components are<br>rectangles with horizontal and vertical sides and with module $m$.