2010/02/19 by Jim Agler, Agler, Jim, John E. McCarthy +3
Mathematics · #Holomorphic and Operator Theory #Numerical methods in inverse problems #Analytic and geometric function theory
paper · pdf · doi:10.48550/arxiv.1002.3727
If \ph is an analytic function bounded by 1 on the bidisk \D2 and τ∈\tb is a point at which \ph has an angular gradient ∇\ph(τ) then ∇\ph(\la) → ∇\ph(τ) as \la→τ nontangentially in \D2. This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For \ph as above, if τ∈\tb is such that the \liminf of (1-|\ph(\la)|)/(1-‖\la‖) as \la→τ is finite then the directional derivative D-\de\ph(τ) exists for all appropriate directions \de∈\C2. Moreover, one can associate with \ph 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.