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A SCHWARZ LEMMA FOR THE SYMMETRIZED BIDISC

Published online by Cambridge University Press:  19 April 2001

J. AGLER  and
N. J. YOUNG
J. AGLER
Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
N. J. YOUNG
Affiliation:
Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU
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Abstract

Let φ be an analytic function from [ ] to the symmetrized bidisc

(formula here)

We show that if φ(0) = (0,0) and φ(λ) = (s, p) in the interior of Γ, then

(formula here)

Moreover, the inequality is sharp: we give an explicit formula for a suitable φ 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 Γ

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 33 , Issue 2 , March 2001 , pp. 175 - 186
Copyright
© The London Mathematical Society 2001

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Footnotes

J. Agler's work is supported by an NSF grant in Modern Analysis. This work was also supported by NATO Collaborative grant CRG 971129.