# A Caratheodory theorem for the bidisk via Hilbert space methods - Mathematics > Complex Variables

A Caratheodory theorem for the bidisk via Hilbert space methods - Mathematics > Complex Variables - Download this document for free, or read online. Document in PDF available to download.

Abstract: If $\ph$ is an analytic function bounded by 1 on the bidisk $\D^2$ and$\tau\in\tb$ is a point at which $\ph$ has an angular gradient$ abla\ph\tau$ then $ abla\ph\la \to abla\ph\tau$ as $\la\to\tau$nontangentially in $\D^2$. This is an analog for the bidisk of a classicaltheorem of Carath\-eodory for the disk.For $\ph$ as above, if $\tau\in\tb$ is such that the $\liminf$ of$1-|\ph\la|-1-\|\la\|$ as $\la\to\tau$ is finite then the directionalderivative $D {-\de}\ph\tau$ exists for all appropriate directions$\de\in\C^2$. Moreover, one can associate with $\ph$ and $\tau$ an analyticfunction $h$ in the Pick class such that the value of the directionalderivative can be expressed in terms of $h$.

Author: ** Jim Agler, John E. McCarthy, Nicholas J. Young**

Source: https://arxiv.org/