Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8805Research ArticleWeighted Inequalities for Quasilinear Integral Operators on the Cone of Monotone FunctionsShambilovaG EMathematical Analysis and Functiona Theory Departmentshambilova@mail.ruPeoples’ Friendship University of Russia150420134334408092016Copyright © 2013,2013Criteria for the Hardy-type inequalities with quasi-linear operators on the cones of monotone functions on the semiaxis are obtained. We study the problem of ﬁnding necessary and suﬃcient conditions for the weighted Hardy-type inequalities for the quasi-linear operators on the cone of monotone functions. To this end we choose a composition of power type integral operations and investigate the characterization problem on its boundedness in the weighted Lebesgue (quasi) norms on the cones of non-negative monotone decrasing functions on the real semiaxis. The main method of the solution of the problem is the reduction method which allows to reduce the inequality on the cones of monotone functions to the corresponding inequalities on the cones of arbitrary non-negative functions, which adopt equivalent description in terms of the boundedness appropriated functionals depending on ingredients of the initial problem. As usual we obtain equivalence of the functionals and the best constants involving into initial inequalities, where the mulpiple constants of equivalence depend only of the parameters of summation. Unlike the initial problems of this area we study multiparametrical case increasing the number of weight functions and summation parameters. This case is new for the weighted inequalities on the cones of monotone functions.Hardy inequalityweighted Lebesgue spacequasi-linear operatorcones of monotone functionsboundednessнеравенство Хардивесовое пространство Лебегаквазилинейный операторконус монотонных функцийограниченность