  We provide some new estimates for Bellman type functions for the dyadic
maximal opeator on $R^n$ 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 different m easures related to the
weight we give new estimates for the above Bellman type functions.
