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A Schwarz Lemma for the symmetrized bidisc

Lookup NU author(s): Professor Jim Agler, Professor Nicholas Young


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Let cp be an analytic function from ID to the symmetrized bidisc Gamma =(def)((lambda (1) +lambda (2), lambda (1)lambda (2)) : \ lambda (1)\ less than or equal to 1, \ lambda (2)\ less than or equal to 1). We show that if phi (0) = (0, 0) and phi(lambda) = (s, p) in the interior of Gamma, then 2 \s - p (s) over bar \+\s(2)-4p \ /4-\s \ (2) less than or equal to \ lambda \. Moreover, the inequality is sharp: we give an explicit formula for a suitable cp in the event that the inequality holds with equality. We show further that the inverse hyperbolic tangent of the left-hand side of the inequality is equal to both the Caratheodory distance and the Kobayashi distance from (0,0) to (s, p) in int Gamma.

Publication metadata

Author(s): Agler J, Young NJ

Publication type: Article

Publication status: Published

Journal: Bulletin of the London Mathematical Society

Year: 2001

Volume: 33

Issue: 2

Pages: 175-186

Print publication date: 01/03/2001

ISSN (print): 0024-6093

ISSN (electronic): 1469-2120

Publisher: Oxford University Press


DOI: 10.1112/blms/33.2.175


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