We generalize several integral inequalities for analytic functions on the open unit polydisc

<

. It is shown that if a holomorphic function on

belongs to the mixed norm space

, where

,j=1,...,n, are admissible weights, then all weighted derivations of order

(with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q

and when the order is equal to one. The equivalence of these conditions is given for all p, q

if

>

, 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.