Facial behaviour of analytic functions on the bidisc

We prove that if φ is an analytic function bounded by 1 on the bidisc D² and τ is a point in a face of D² at which φ satisfies Carathéodory's condition, then both φ and the angular gradient ∇φ exist and are constant on the face. Moreover, the class of all φ with prescribed φ(τ) and ∇φ(τ) can be parametrized in terms of a function in the two-variable Pick class. As an application we solve an interpolation problem with nodes that lie on the faces of the bidisc.