## On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation

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- by David Kalaj and Miroslav Pavlović PDF
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## Abstract:

Let $\mathcal {QC}(K,g)$ be a family of $K$-quasiconformal mappings of the open unit disk onto itself satisfying the PDE $\Delta w =g$, $g\in C(\overline {\mathbb {U}})$, $w(0)=0$. It is proved that $\mathcal {QC}(K,g)$ is a uniformly Lipschitz family. Moreover, if $|g|_\infty$ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as $K \to 1$ and $|g|_\infty \to 0$, so $w\in \mathcal {QC}(K,g)$ behaves almost like a rotation for sufficiently small $K$ and $|g|_\infty$.## References

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## Additional Information

**David Kalaj**- Affiliation: Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica, Montenegro
- MR Author ID: 689421
- Email: davidkalaj@gmail.com
**Miroslav Pavlović**- Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia
- Email: pavlovic@matf.bg.ac.rs
- Received by editor(s): May 7, 2008
- Received by editor(s) in revised form: April 12, 2009
- Published electronically: March 23, 2011
- © Copyright 2011 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**363**(2011), 4043-4061 - MSC (2010): Primary 30C62
- DOI: https://doi.org/10.1090/S0002-9947-2011-05081-6
- MathSciNet review: 2792979