Abstract
In this short note, we show that the tetrablock is a \({\mathbb{C}}\)-convex domain. In the proof of this fact, a new class of (\({\mathbb{C}}\)-convex) domains is studied. The domains are natural candidates to study on them the behavior of holomorphically invariant functions.
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References
A. A. Abouhajar, M. C. White, and N. J. Young, A Schwarz lemma for a domain related to mu-synthesis, J. Geom. Analysis, 17 (2007), 717–750.
Agler J., Young N. J (2004) The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14: 375–403
M. Andersson, M. Passare, and R. Sigurdsson, Complex convexity and analytic functionals, Birkhäuser, Basel-Boston-Berlin, 2004.
T. Bhattacharyya, Operator theory on the tetrablock, preprint, 2012, http://arxiv.org/abs/1207.3395.
Costara v (2004) The symmetrized bidisc and Lempert’s theorem, Bull. London Math. Soc. 36: 656–662
A. Edigarian, A note on C. Costara’s paper: “The symmetrized bidisc and Lempert’s theorem” [Bull. London Math. Soc. 36 (2004), 656–662], Ann. Polon. Math. 83 (2004), 189–191.
A. Edigarian, Ł. Kosiński, and W. Zwonek, The Lempert Theorem and the tetrablock, J. Geom. Analysis, to appear.
L. Hörmander, Notions of Convexity, Birkhäuser, Basel-Boston-Berlin, 1994.
M. Jarnicki and P. Pflug, Invariant distances and metrics in complex analysis, de Gruyter, Berlin-New York, 1993.
Kosiński Ł (2011) Geometry of quasi-circular domains and applications to tetrablock, Proc. Amer. Math Soc., 139: 559–569
Lempert L (1981) La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109: 427–474
L. Lempert, Intrinsic distances and holomorphic retracts, in Complex analysis and applications 81 (Varna, 1981), 341–364, Publ. House Bulgar. Acad. Sci., Sofia, 1984.
N. Nikolov, P. Pflug, and W. Zwonek, An example of a bounded \({\mathbb{C}}\)-convex domain which is not biholomorphic to a convex domain, Math. Scand. 102 (2008), 149–155
Pflug P., Zwonek W (2012) Exhausting domains of the symmetrized bidisc, Ark. Mat. 50: 397–402
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The work is partially supported by the grant of the Polish National Science Centre no. UMO-2011/03/B/ST1/04758.
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Zwonek, W. Geometric properties of the tetrablock. Arch. Math. 100, 159–165 (2013). https://doi.org/10.1007/s00013-012-0479-7
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DOI: https://doi.org/10.1007/s00013-012-0479-7