A Schwarz lemma for the symmetrized bidisc

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 o f Gamma, then 2 \s - p (s) over bar \+\s(2)-4p \ /4-\s \ (2) less than or equal to \ lamb da \. Moreover, the inequality is sharp: we give an explicit formula for a suitab le 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 inequalit y is equal to both the Caratheodory distance and the Kobayashi distance fro m (0,0) to (s, p) in int Gamma.