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I am studying conformal mappings of multiply connected domains. Most of the results that I can reach concern existence and uniqueness of such mappings, whereas I can not find anything satisfactory concerning the constructive side of the problem. More specifically I have the following problem: consider a domain which is formed by deleting several non-overlapping disks from the unit disk. The centers of the deleted disks are on the positive real axis and are explicitly known as well as their radii. We know that we can map this domain (uniquely) conformally to the unit disk less some closed intervals also contained in the positive real axis. The question is how are these intervals situated. Can we effectively find their endpoints? If not, can we estimate their position by some explicit expression involving the data (centers and radii of the deleted disks)?

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There is a considerable literature on effective conformal mapping. For a not-too-old survey see Breakthrough in Conformal Mapping, by John Case.

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