Quasisymmetric mappings in b-metric spaces

Considering quasisymmetric mappings between b-metric spaces we have found a new estimation for the ratio of diameters of two subsets which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. The condition under which the image of a b-metric space under quasisymmetry is also a b-metric space is established. Moreover, the latter question is investigated for additive metric spaces.