A Caratheodory theorem for the bidisk via Hilbert space methods

If $\text{ph}$ is an analytic function bounded by 1 on the bidisk ${\text{D}}^2$ and $\tau \in \text{tb}$ is a point at which $\text{ph}$ has an angular gradient $\mathrm{\nabla }\text{ph}(\tau )$ then $\mathrm{\nabla }\text{ph}(\text{la})\to \mathrm{\nabla }\text{ph}(\tau )$ as $\text{la}\to \tau$ nontangentially in ${\text{D}}^2$. This is an analog for the bidisk of a classical theorem of Carath'eodory for the disk. For $\text{ph}$ as above, if $\tau \in \text{tb}$ is such that the $\mathrm{lim\hspace{0.17em}inf}$ of $(1-\mathrm{\mid }\text{ph}(\text{la})\mathrm{\mid })\mathrm{/}(1-\mathrm{\parallel }\text{la}\mathrm{\parallel })$ as $\text{la}\to \tau$ is finite then the directional derivative $D_{-\text{de}}\text{ph}(\tau )$ exists for all appropriate directions $\text{de}\in {\text{C}}^2$. Moreover, one can associate with $\text{ph}$ and $\tau$ an analytic function $h$ in the Pick class such that the value of the directional derivative can be expressed in terms of $h$.