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.

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).

In the homogenization theory, there are many examples where the effective conductivities of composite structures are power means of the local conductivities. The main aim of this paper is to initiate research concerning geometric construction of some power means of three or more variables. We contribute by giving methods for the geometric construction of the harmonic mean $ P_{-1} $ and the arithmetic ...

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

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.

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 ...

The aim is to put new light on the single ladder problem (SLP). Some new methods for finding complete integer solutions to the corresponding quartic equation z 4 −2L z 3 +( L 2 − a 2 − b 2 ) z 2 +2L a 2 z− L 2 a 2 =0 z4−2Lz3+(L2−a2−b2)z2+2La2z−L2a2=0 are developed. For the case L≥ L min L≥Lmin , these methods imply a complete parametric representation for integer ...

We obtain some further refinements of Hardy-type inequalities via superqudraticity technique. Our results both unify and further generalize several results on refinements of Hardy-type inequalities in the literature.

In this paper we prove some essential complements of the paper (J. Inequal. Appl. 2019:171, 2019) on the same theme. We prove some new Fourier inequalities in the case of the Lorentz–Zygmund function spaces L q,r (logL ) α Lq,r(log⁡L)α involved and in the case with an unbounded orthonormal system. More exactly, in this paper we prove and discuss some new Fourier inequalities of this type for ...