On analytic maps of plane domains (Q1103062)

Let D be an n-ply connected plane domain bounded by n (\(\geq 3)\) analytic curves. Let f(z) be a boundary preserving analytic map of D, whose image domain \(\Delta\) consists of more than two boundary components. Such an f can be extended over the doubled surface \(\hat D\) and the extended maps \(\hat f(z)\) is an analytic map of the closed Riemann surface \(\hat D\) onto the doubled surface \(\Delta\). Hence, by the deFranchis theorem, the number of equivalence classes (suitably defined) of boundary preserving map of D is finite. In this paper, the authors prove that this number is bounded by \((n-2)2^{4n-6}\). To do so, they obtain a sufficient condition that two boundary preserving maps f and g of D satisfy \(f(z)=\phi(g(z))\) for a conformal map \(\phi\) of g(D) onto f(D). Let \(A_ n\) be the maximal number of equivalence classes of boundary preserving analytic maps of n-ply connected plane domains bounded by n (\(\geq 3)\) analytic curves. Then they prove that \(A_ 3=6\) and \(A_ 4=12\).