Uniqueness theorems for analytic vector-valued functions

Using the Berezin transformation, we give a multidimensional analog of a uniqueness theorem of N.Nikolski concerning distance functions and subspaces of a Hilbert space of analytic functions. Then, we establish some uniqueness properties drawing connections between two analytic $X$-valued functions $F$ and $G$ that satisfy $\|F(z)\|\equiv\|G(z)\|,\,\forall z\in\Omega$, where $X$ is a Banach space and $\Omega$ a connected domain in $\mathbb C^n$. The particular case where $X=\ell_n^p$ and $\Omega=\mathbb D=\{z\in\mathbb C\,:\,|z|<1\,\}$ will lead us to the notion of flexible and inflexible functions. We give a complete description of these functions when $p=+\infty,\,n\in\mathbb N^*$ and when $n=2,\,1\le p\le+\infty$.