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Apr 27, 2014 at 18:37 comment added Alexandre Eremenko What you are saying is plausible, but the details should be checked.
Apr 27, 2014 at 14:59 comment added Moishe Kohan I see: This is a nice argument for connectivity (I think, even path connectivity). Can it be promoted to contractibility in the case of domains of finite topological type? The key issue, as I see it, is lack of a canonical quasiconformal extension to the entire sphere. Maybe one can use extension by a harmonic quasiconformal map (if boundary of the image is nice enough) or Ahlfors-Beurling.
Apr 27, 2014 at 14:18 history answered Alexandre Eremenko CC BY-SA 3.0