The quasimöbius property on small circles and quasiconformality

We prove that every mapping, without requiring its injectivity or continuity, of a domain of the extended plane which is $\omega$-quasimöbius on sufficiently small circles is locally quasiconformal in this domain with an upper bound on the quasiconformality coefficient depending only on $\omega$. We obtain a similar result for the $\eta$-quasisymmetric mappings on small circles (in the Euclidean and chordal metrics), as well as for the mappings satisfying the local Möbius midpoint condition.