Local properties of quasihyperbolic and freely quasiconformal mappings

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