A discrete Hardy-type inequality ( ∑∞ n=1( ∑n k=1dn,kak)un) ≤ C( ∑∞ n=1 a p nvn) is considered for a positive “kernel” d = {dn,k}, n,k ∈ Z+, and p ≤ q. For kernels of product type some scales of weight characterizations of the inequality are proved with the corresponding estimates of the best constant C. A sufficient condition for the inequality to hold in the general case is proved and this co...

We establish necessary and sufficient conditions for the one-dimensional differential Hardy inequality to hold, including the overdetermined case. The solution is given in terms different from those of the known results. Moreover, the least constant for this inequality is estimated.

We establish a Maz’ya type capacitary inequality which resolves special case of conjecture by David R. Adams. As consequence, we obtain several equivalent norms for Choquet integrals associated to Bessel or Riesz capacities. This enables us bounds the Hardy–Littlewood maximal function in sublinear setting.

In this paper we derive new inequalities involving the generalized Hardy operator. The obtained results known We also get classical Hardy–Hilbert inequality.