If φ is an analytic function bounded by 1 on the bidisk D 2 and τ∈∂(D 2 ) is a point at which φ has an angular gradient ∇φ(τ) then ∇φ(λ)→∇φ(τ) as λ → τ nontangentially in D 2 . This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For φ as above, if τ∈∂(D 2 ) is such that the lim inf of (1−|φ(λ)|)/(1−∥λ∥) as λ → τ is finite then the directional derivative D−δφ(τ) exists for all appropriate directions δ∈C 2 . Moreover, one can associate with φ and τ an analytic function h in the Pick class such that the value of the directional derivative can be expressed in terms of h.
Agler, Jim; McCarthy, John E.; and Young, N J., "A Carathéodory theorem for the bidisk via Hilbert space methods" (2012). Mathematics Faculty Publications. 8.