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