Moduli of ring domains obtained by a conformal welding (Q1392769)

Let \(Q=\{x+ iy:0<x< 1,0<y<1\}\) be the unit square in the complex plane, with top edge \(L_+\) and bottom edge \(L_-\). Suppose that we are given an increasing function \(\varphi\) from \(L_+\) to \(L_-\). Let \(G\) be a ring domain in the plane and let \(C\) be a Jordan arc joining the boundary components of \(G\). Let \(f\) be a continuous mapping of \(Q\cup L_+ \cup L_-\) onto \(G\), such that \(f\) is conformal in \(Q,f\circ \varphi=f\) on \(L_+\) and \(f(L_+)= f(L_-) =C\). Then \((G,C,f)\) is called a conformal welding obtained by \(\varphi\) and \(\varphi\) is called a welding function. Now let \(M(G)\) be the modulus of the family of curves in \(G\) which separate the boundary components of \(G\). (The authors define the modulus for an annulus of radii \(r_1\) (inner) and \(r_2\) (outer) to be \(\log (r_2/r_1.)\) The authors are interested in the set \(M_\varphi= \{M(G): (G,C,f)\) is a conformal welding obtained by \(\varphi\}\). Oikawa proved several unpublished results on possible values of \(M_\varphi\). The authors utilize quasiconformal mappings to prove: 1) There exists a welding function \(\varphi\) and \(\varepsilon>0\) such that \(M_\varphi \supset (0,\varepsilon)\). 2) For any nontrivial \(\varphi\), there is an \(m<2\pi\) such that \(M_\varphi \subset (0,m]\).