Distance Ratio Metric
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The basic distortion results about quasiconformal mappings such as the Schwarz lemma and the Gehring–Osgood theorem say that these mappings are Hölder continuous with respect to the hyperbolic and the quasihyperbolic metric respectively. In this chapter we analyze the modulus of continuity in the case of the distance ratio metric. The natural question is to find Lipschitz constants for this metric under Möbius transformations or arbitrary holomorphic mappings. The domains we work with here are the unit ball, the punctured ball, and the upper half space.
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