We prove necessary and sufficient integral conditions involving extremal distance for a conformal mapping of the unit disk to belong Hardy or weighted Bergman spaces. also give characterizations number simply connected domain in terms distance.

For a wide range of pairs mixed norm spaces such that one space is contained in another, we characterize all cases when contractive inequalities hold. In particular, this yields results for many weighted Bergman spaces. Some type are motivated by their applications Number Theory and Mathematical Physics.