Local properties of quasihyperbolic and freely quasiconformal mappings

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\subset E$ and $D'\subset E'$ are domains, and that $f: D\to D'$ is a homeomorphism. In this paper, we prove that if there exists some constant $M>1$ (resp. some homeomorphism $\phi$) such that for all $x\in D$, $f: B(x,d_D(x))\to f(B(x,d_D(x)))$ is $M$-QH (resp. $\phi$-FQC), then $f$ is $M_1$-QH with $M_1=M_1(M)$ (resp. $\phi_1$-FQC with $\phi_1=\phi_1(\phi)$). We apply our results to establish, in terms of the $j_D$ metric, a sufficient condition for a homeomorphism to be FQC.