Suppose that (Formula presented.) and (Formula presented.) denote real Banach spaces with dimension at least 2 and that (Formula presented.) and (Formula presented.) are domains. Let (Formula presented.) be a homeomorphism with (Formula presented.). We say that a homeomorphism (Formula presented.) is (Formula presented.)-FQC if for every subdomain (Formula presented.), we have (Formula presented.) holds for all (Formula presented.). In this paper, we establish, in terms of the (Formula presented.) metric, a necessary and sufficient condition for a homeomorphism (Formula presented.) to be FQC. Moreover, we give, in terms of the (Formula presented.) metric, a sufficient condition for a homeomorphism (Formula presented.) to be FQC. On the other hand, we show that this condition is not necessary.