On some sharp theorems on distance function in Hardy type, Bergman type and Herz type analytic classes

We present some new sharp estimates concerning distance function in some new mixed norm and Lizorkin–Triebel type spaces in the unit ball.This leads at the same time to direct generalizations of our recent results on extremal problems in such Bergman type spaces. In addition new sharp results in this direction in Hardy-Morrey, and some new weighted Hardy classes in the unit ball and mixed norm Hardy type spaces and Herz type spaces in polyball will be provided and discussed.