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Riemann mapping is nuclear weapons given the relatively crude inequalities of the original questions. Perhaps the original inquiry helps amplify a trichotomy among true'',obvious'', and ``why exactly is this true?''.
If we allow the Riemann mapping theorem, take a conformal map from the (punctured) open unit disk minus (0,0) onto the complement of K. The images in the plane of the circles centered at (0,0) of radius 1/n will do.
