Computational Methods and Function Theory 8 (2008), No. 1, 225--242
Copyright Heldermann Verlag 2008
Steven R. Bell
bell@math.purdue.edu , Purdue University, Mathematics Department, West Lafayette, IN 47907, U.S.A.
Faisal Kaleem
kaleem@ulm.edu , University of Louisiana, Monroe, Department of Mathematics, Monroe, LA 71209, U.S.A.
Given a bounded $n$-connected domain $\Omega$ in the plane bounded by $n$~non-intersecting Jordan curves and given one point $b_j$ on each boundary curve, L.~Bieberbach proved that there exists a proper holomorphic mapping~$f$ of $\Omega$ onto the unit disc that is an $n$-to-one branched covering with the properties: $f$ extends continuously to the boundary and maps each boundary curve one-to-one onto the unit circle, and $f$ maps each given point $b_j$ on the boundary to the point $1$ in the unit circle. We shall modify a proof by H.~Grunsky of Bieberbach's result to show that there is a rational function of $2n+2$ complex variables that generates all of these maps. In fact, we show that there are two Ahlfors maps $f_1$ and $f_2$ associated with the domain such that any such mapping is given by a fixed linear fractional transformation mapping the right half plane to the unit disc composed with $c\,R + i\,C$, where $R$ is a rational function of the $2n+2$ functions $f_1(z),f_2(z)$ and $f_1(b_1),f_2(b_1),\dots,f_1(b_n),f_2(b_n)$, and where $c$ and $C$ are arbitrary real constants subject to the condition $c>0$. We also show how to generate \emph{all} the proper holomorphic mappings to the unit disc via the rational function~$R$.
Keywords: Poisson kernel, Grunsky maps.
MSC 2000: 30C35.
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