Summary (translated from the Russian): "We consider modular inequalities and inequalities for norms of Hardy type operators on the cone of positive functions, and also on the cone Omega of positive decreasing functions in a weighted Orlicz space with a common weight and a common Young function. We establish a reduction theorem for the norm of the Hardy operator on the cone Omega. We show that it is equivalent to the norm of a modified operator on the cone of all positive functions in this space. We establish that the modified operator is a generalized Hardy type operator. We show the equivalence of modular inequalities on the cone Omega and modified modular inequalities on the cone of all positive functions in the Orlicz space. We obtain a criterion for the validity of these inequalities in general weighted Orlicz spaces and we give its concrete definition for weighted Lebesgue spaces."