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Bulletin of the London Mathematical Society Advance Access originally published online on March 22, 2009
Bulletin of the London Mathematical Society 2009 41(3):506-514; doi:10.1112/blms/bdp022
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© 2009 London Mathematical Society

Schwarz lemma for the tetrablock

Armen Edigarian and Wlodzimierz Zwonek

Institute of Mathematics
Jagiellonian University
Lojasiewicza 6
30-348 Kraków
Poland
Wlodzimierz.Zwonek@im.uj.edu.pl

Received 13 March 2008. Revision received 30 October 2008.

We describe all complex geodesics in the tetrablock passing through the origin thus obtaining the form of all extremals in the Schwarz lemma for the tetrablock. Some other extremals for the Lempert function and geodesics are also given. The paper may be seen as a continuation of the results from Abouhajar et al. [‘A Schwarz lemma for a domain related to mu-synthesis’, J. Geom. Anal. 17 (2007) 717–750]. The proofs rely on a necessary form of complex geodesics in general domains which is also proven in the paper.

2000 Mathematics Subject Classification 32F45 (primary).

The paper was partially supported by the Research Grant of the Polish Ministry of Science and Higher Education No. 1 PO3A 005 28.


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