In this paper, we prove that 1/|?|2-harmonic quasiconformal mapping is
bi-Lipschitz continuous with respect to quasihyperbolic metric on every
proper domain of C\{0}. Hence, it is hyperbolic quasi-isometry in every
simply connected domain on C\{0}, which generalized the result obtained in
[14]. Meanwhile, the equivalent moduli of continuity for 1/|?|2-harmonic
quasiregular mapping are discussed as a by-product.
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<abstract><p>In this paper, three equivalent conditions of $ \rho $-harmonic Teichmüller mapping are given firstly. As an application, we investigate the relationship between a $ \rho $-harmonic Teichmüller mapping and its associated holomorphic quadratic differential and obtain a relatively simple method to prove Theorem 2.1 in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. Furthermore, the representation theorem of $ 1/|\omega|^{2} $-harmonic Teichmüller mappings is given as a by-product. Our results extend the corresponding researches of harmonic Teichmüller mappings.</p></abstract>
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