Isoperimetric inequality for the polydisk

Let n is an element of N. For k is an element of {1,......, n} let Omega(k) subset of C be a simply connected domain with a rectifiable boundary. Let Omega(n) = Pi(n)(k=1) subset of C-n be a generalized polydisk with distinguished boundary partial derivative Omega(n) = Pi(n)(k=1) partial derivative Omega(k). Let E (r) (Omega (n) ) be the holomorphic Smirnov class on Omega (n) with index r. We show that the generalized isoperimetric inequality integral vertical bar f(1)vertical bar(p) vertical bar f(2)vertical bar(q) dv <= 1/(4 pi)(n) integral(partial derivative Omega n) vertical bar f(1)vertical bar (p)ds integral(partial derivative Omega n) vertical bar f(2)vertical bar(q) dS, holds for arbitrary f(1) is an element of E-p (Omega(n)) and f(2) is an element of E-q (Omega(n)), where 0 < p, q < infinity. We also determine necessary and sufficient conditions for equality.