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.

我们研究拟共形映射的平均半径增长函数。^{我们在 ℝ n}中给出了一个新的拟共形映射子类，对于n ≥ 2，称为有界可积参数化映射，简称 BIP 映射。它们具有以下特性：Zorich 变换对每个切片的限制在L ^{n /( n -1)}中具有均匀有界导数。对于 BIP 地图，平均半径函数的对数变换是 bi-Lipschitz。然后，我们将结果应用于具有简单无穷小空间的 BIP 映射，通过证明其 Zorich 变换是双 Lipschitz 映射来表明渐近表示确实是拟共形的。