We provide some new estimates for Bellman type functions for the dyadic maximal operator Rn and of maximal operators on martingales related to weighted spaces. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we introduce certain conditions on the weight that imply estimate for the maximal operator on the corresponding weighted space. Also using a well-known estimate for the maximal operator by a double maximal operators on deferent measures related to the weight we give new estimates for the above Bellman type functions.
