Freely quasiconformal maps and distance ratio metric

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least $2$ and that $D\subset E$ and $D'\subset E'$ are domains. In this paper, we establish, in terms of the $j_D$ metric, a necessary and sufficient condition for the homeomorphism $f: E \to E'$ to be FQC. Moreover, we give, in terms of the $j_D$ metric, a sufficient condition for the homeomorphism $f: D\to D'$ to be FQC. On the other hand, we show that this condition is not necessary.