A Schwarz Lemma for the Symmetrized Bidisc

Let φ be an analytic function from D to the symmetrized bidisc Γ = def { ( λ 1 + λ 2 , λ 1 λ 2 ) : ∣ λ 1 ∣ ⩽ 1 , ∣ λ 2 ⩽ 1 ∣ } . We show that if φ(0) = (0,0) and φ(λ) = (s, p) in the interior of Γ, then 2 ∣ s − p s ¯ ∣ + ∣ s 2 − 4 p ∣ 4 − ∣ s ∣ 2 ⩽ ∣ λ ∣ . 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