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 ...

In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin–Fourier (Walsh–Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh–Fourier series.

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.

Some new multidimensional Hardy-type inequalites are proved and discussed. The cases with bilinear and iterated operators are considered and some equivalence theorems are proved.

A number of classical inequalities and convergence results related to Fourier coefficients with respect to unbounded orthogonal systems are generalized and complemented. All results are given in the case of Lorentz–Zygmund spaces.