On the volumes of images of holomorphic mappings in $\textbf {C}^ n$

Abstract:

Let $f$ be a holomorphic map from the unit ball in complex $n$-space to complex $n$-space. We establish a lower bound for the volume, taken without multiplicity, of the image of $f$. The estimate is in terms of the boundary values of $f$. This generalizes some known results in one complex variable. The proof uses the methods of uniform algebras.