On the subinvariance of inner uniform domains in Banach spaces

Suppose that E and \(E'\) $E^{\prime }$ denote real Banach spaces with dimension at least 2, that \(G\subsetneq E\) $G⊊E$ and \(G'\subsetneq E'\) $G^{\prime }⊊E^{\prime }$ are domains, and that \(f: G\rightarrow G'\) $f:G\to G^{\prime }$ is a quasihyperbolic mapping. It is known that uniform domains have subinvariance property under quasihyperbolic mappings where \(G'\) $G^{\prime }$ is uniform. In this paper, we show that inner uniform domains do not have subinvariance property under quasihyperbolic mappings, and we prove that the image of every uniform subdomain D in G is inner uniform when \(G'\) $G^{\prime }$ is inner uniform.