We generalize several integral inequalities for analytic functions on the open unit polydisc $U^n={{}z{in}C^n||zj|<1,;j=1,...,n{}}$. It is shown that if a holomorphic function on $U^n$ belongs to the mixed norm space $A_{vec{omega}}^{p,q}(U^n)$, where ${omega}_j(cdot)$,j=1,...,n, are admissible weights, then all weighted derivations of order $|k|$ (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ${in};[1,;{infty})$ and when the order is equal to one. The equivalence of these conditions is given for all p, q ${in};(0,;{infty})$ if ${omega}_j(z_j)=(1-|z_j|^2)^{{alpha}j},;{alpha}_j>-1$, j=1,...,n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577-587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775-782.