On the mean radius of quasiconformal mappings

We study the mean radius growth function for quasiconformal mappings. We give a new sub-class of quasiconformal mappings in ℝⁿ, for n ≥ 2, called bounded integrable parameterization mappings, or BIP maps for short. These have the property that the restriction of the Zorich transform to each slice has uniformly bounded derivative in L^n/(n−1). For BIP maps, the logarithmic transform of the mean radius function is bi-Lipschitz. We then apply our result to BIP maps with simple infinitesimal spaces to show that the asymptotic representation is indeed quasiconformal by showing that its Zorich transform is a bi-Lipschitz map.