Let
\({n\in\mathbb{N}}\). For
\({k\in\{1,\dots,n\}}\) let
\({\Omega_k\subset \mathbb{C}}\) be a simply connected domain with a rectifiable boundary. Let
\({\Omega^n=\prod_{k=1}^n\Omega_k\subset \mathbb{C}^n}\) be a generalized polydisk with distinguished boundary
\({\partial\Omega^n=\prod_{k=1}^n\partial\Omega_k}\). Let
E r (Ω
n ) be the holomorphic Smirnov class on Ω
n with index
r. We show that the generalized isoperimetric inequality
$ \int\limits_{\Omega^n} |f_1|^p|f_2|^qdV\le \frac{1}{(4\pi)^n}\int\limits_{\partial \Omega^n}|f_1|^pdS \int\limits_{\partial \Omega^n} |f_2|^qdS, $
holds for arbitrary
\({f_1\in E^p(\Omega^n)}\) and
\({f_2\in E^q(\Omega^n)}\), where 0 <
p,
q < ∞. We also determine necessary and sufficient conditions for equality.