Relatively quasim\"{o}bius mappings in Banach spaces.

In this paper, we explore relatively quasimöbius invariance of \varphi-uniform domains and natural domains. Firstly, we prove that the control function of relatively quasimöbius mappings can be chosen in a power form. Applying this observation and a deformed cross–ratio introduced by Bonk and Kleiner, we next show that relatively quasimöbius mappings are coarsely bilipschitz in the distance ratio metric. Combined with the assumption that the mapping is coarsely bilipschitz in the quasihyperbolic metric, we establish relatively quasimöbius invariance of \varphi-uniform domains and natural domains. As a by-product, we also obtain a similar result for uniform domains which provides a new method to answer a question posed by Väisälä. [ABSTRACT FROM AUTHOR]