Some integral estimates on the cones of functions with the monotonicity conditions

In this paper we obtain estimates for the integrals of monotone functions arising in the study of the covering of various cones of functions with monotonicity conditions. We apply the method of covering of the cones with the help of generalized Hardy operator. Sharp conditions are found on the kernels of representations for the validity of given estimates on the cones. The proofs are based on the reduction of integral estimates on the cones of monotone functions to ones on the family of characteristic functions of intervals. The obtained results can be used in finding the condition for the mutual covering of cones associated with decreasing rearrangement of the generalized Bessel and Riesz potentials.