A Schwarz Lemma for the symmetrised bidisc

Jim Agler and Nicholas Young

Keywords

Caratheodory distance, Kobayashi distance, mu-synthesis

Status

Appeared in the Bulletin of the London Mathematical Society, Volume 33, March 2001, pages 175-186.

Abstract

Let $\varphi$ be an analytic function from $\Bbb D$ to the symmetrized bidisc $$ \Gamma \stackrel{\rm def}{=} \{(\lambda_1 + \lambda_2, \lambda_1\lambda_2): |\lambda_1|\le 1, |\lambda_2|\le 1\}. $$ We show that if $\varphi(0)=(0, 0)$ and $\varphi(\lambda) =(s, p)$ in the interior of $\Gamma$ then

\frac{2|s - p \overline s| + |s^2 - 4p|}{4-|s|^2} \leq |\lambda|.

Moreover the inequality is sharp: we give an explicit formula for a suitable $\varphi$ in the event that the inequality holds with equality. We show further that the left hand side of the inequality is equal to the hyperbolic tangent of both the Caratheodory distance and the Kobayashi distance from $(0,0)$ to $(s,p)$ in int $\Gamma$.

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

Nicholas Young
06 May 1999.