New weighted Hardy-type inequalities for monotone functions

Abstract

The famous Hardy inequality was formulated in 1920 and finally proved in 1925. Since then, this inequality has been greatly developed. The first development was related to the consideration of more general weights. The next step was to use more general operators with different kernels instead of the Hardy operator. At present, there are many works devoted to Hardy-type inequalities with iterated operators. Motivated by important applications, all these generalizations of the Hardy inequality are studied not only on the cone of non-negative functions but also on the cone of monotone non-negative functions. In this paper, new Hardy-type inequalities are proved for operators with kernels that satisfy less restrictive conditions than those considered earlier. The presented inequalities have already been characterized for non-negative functions. In this paper, we continue this study but for monotone non-negative functions