Refinements of some limit Hardy-type inequalities are derived and discussed using the concept of superquadracity. We also proved that all three constants appearing in the refined inequalities obtained are sharp. The natural turning point of our refined Hardy inequality is p=2 and for this case we have even equality.

Some new extensions and re finements of Hermite – Hadamard and Fejer type inequalities for functions which are N -quasiconvex are derived and discussed.

The classical Hausdorff-Young and Hardy-Littlewood-Stein inequalities do not hold for p > 2. In this paper we prove that if we restrict to net spaces we can even derive a two-sided estimate for all p > 1. In particular, this result generalizes a recent result by Liflyand E. and Tikhonov S. [7] (MR 2464253).

Some classical inequalities are known also in a more general form of Banach lattice norms and/or in continuous forms (i.e., for ‘continuous’ many functions are involved instead of finite many as in the classical situation). The main aim of this paper is to initiate a more consequent study of classical inequalities in this more general frame. We already here contribute by discussing some results of ...

In this paper, we derive two-sided estimates of the Lebesgue constants for bounded Vilenkin systems, we also present some applications of importance, e.g., we obtain a characterization for the boundedness of a subsequence of partial sums with respect to Vilenkin–Fourier series of H 1 martingales in terms of n's variation. The conditions given in this paper are in a sense necessary and sufficient.

We prove the boundedness of Potential operator in weighted generalized Morrey space in terms of Matuszewska-Orlicz indices of weights and apply this result to the Hemholtz equation in ℝ³ with a free term in such a space. We also give a short overview of some typical situations when Potential type operators arise when solving PDEs.