Two-sided hardy-type inequalities for monotone functions

We consider Hardy-type operators on the cones of monotone functions with general positive σ-finite Borel measure. Some two-sided Hardy-type inequalities are proved for the parameter -∞ < p < ∞. It is pointed out that such equivalences, in particular, imply a new characterization of the discrete Hardy inequality for the (most difficult) case 0 < q < p ≤ 1. © 2010 Taylor & Francis.