  We prove a multiparameter integral inequality for the dyadic maximal operator
which refines the one-parameter inequality that is given by A. Melas in [10]
which in turn is applied for the evaluation of the Bellman function of two
integral variables for this maximal operator. Moreover we find the exact domain
of definition of the related Bellman function of three integral variables and
by using the results connected with the sharpness of this new multiparameter
inequality we give lower bounds of this Bellman function.
