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On the mean radius of quasiconformal mappings

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Abstract

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 Ln/(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.

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Acknowledgement

The authors would like to thank the referees for a number of suggestions which helped improve the exposition of this paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115-2888, USA

    Alastair N. Fletcher

  2. Department of Mathematics and Statistics, Stephen F. Austin State University, Nacogdoches, TX, 75962, USA

    Jacob Pratscher

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  1. Alastair N. Fletcher
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  2. Jacob Pratscher
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Correspondence to Jacob Pratscher.

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Fletcher, A.N., Pratscher, J. On the mean radius of quasiconformal mappings. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2583-8

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  • DOI: https://doi.org/10.1007/s11856-023-2583-8

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