This paper is devoted to investigating the sequence of some linear functionals in the space $BV$ of finite variation functions. We prove that under certain conditions this sequence is bounded. We also prove that this result is sharp. In particular, the obtained results can be used to study convergence of some general Fourier series. Moreover, the obtained conditions seem to be new and useful also for classical orthonormal systems.

In this manuscript a number of general inequalities for isotonic subadditive functionals on a set of positive mappings are proved and applied. In particular, it is pointed out that these inequaliti ...

In this survey, we explain and discuss some recent results concerning the close connection between Carlson type inequalities and interpolation theory. In particular, we point out that a fairly general Carlson type inequality can be used to extend the usefulness of the Gustavsson-Peetre h¢i' interpolation method.

Some Hardy-type integral inequalities in general measure spaces, where the corresponding Hardy operator is replaced by a more general Volterra type integral operator with kernel k(x, y), are considered. The equivalence of such inequalities on the cones of non-negative respective non-increasing functions are established and applied.

We investigate Carlson type inequalities for finite sums, that is, inequalities of the form to hold for some constant C independent of the finite, non-zero set a1,…,am of non-negative numbers. We find constants C which are strictly smaller than the sharp constants in the corresponding infinite series case. Moreover, corresponding results for integrals over bounded intervals are given and a case with any finite number of factors on the right-hand side is proved.

The conditions are investigated, under which for all Lebesgue measurable functions f(x) greater than or equal 0 on a semi-axis R+:=(0, infinity ) with a constant C greater than or equal 0 independe ...

A recently discovered Hardy-Polya type inequality described by a convex function is considered and further developed both in weighted and unweighted cases. Also some corresponding multidimensional and reversed inequalities are pointed out. In particular, some new multidimensional Hardy and Polya-Knopp type inequalities and some new integral inequalities with general integral operators (without additional restrictions on the kernel) are derived.

We give a simpler proof of the formula, due to J.-L. Lions, for the reproducing kernel of the space of harmonic functions on a domain Omegasubset ofR(n) whose boundary values belong to the Sobolev space H-s(partial derivativeOmega), and also obtain generalizations of this formula when instead of harmonic functions one considers functions annihilated by a given elliptic partial differential operator. Further, we compute the reproducing kernels explicitly in several examples, which leads to an occurrence of new special functions. Some spaces of caloric functions are also briefly considered.

In this paper we consider inequalities of the form, WhereMis a mean. The main results of the paper offer sufficient conditions onMso that the above inequality holds with a finite constantC. The results obtained extend Hardy's and Carleman's classical inequalities together with their various generalisations in a new dirction.

In this article we discuss lattice convexity and concavity of Calderón-Lozanovskii space $E_\varphi$, generated by a quasi-Banach space $E$ and an increasing Orlicz function $\varphi$. We give estimations of convexity and concavity indices of $E_\varphi$ in terms of Matuszewska-Orlicz indices of $\varphi$ as well as convexity and concavity indices of $E$. In the case when $E_\varphi$ is a rearrangement invariant space we also provide some estimations of its Boyd indices. As corollaries we obtain some necessary and sufficient conditions for normability of $E_\varphi$, and conditions on its nontrivial type and cotype in the case when $E_\varphi$ is a Banach space. We apply these results to Orlicz-Lorentz spaces receiving estimations, and in some cases the exact values of their convexity, concavity and Boyd indices.

In this paper we prove a new refinement of the weighted arithmetic-geometric mean inequality and apply this result in obtaining a sharpened version of the weighted Carleman's inequality.

In this paper we prove a strengthened general inequality of the Hardy–Knopp type and also derive its dual inequality. Furthermore, we apply the obtained results to unify the strengthened classical Hardy and Pólya–Knopp's inequalities deriving them as special cases of the obtained general relations. We discuss Pólya–Knopp's inequality, compare it with Levin–Cochran–Lee's inequalities and point out that these results are mutually equivalent. Finally, we also point out a reversed Pólya–Knopp type inequality.

J. Arazy [1] pointed out that there is a similarity between functions defined on the torus and infinite matrices. In this paper we discuss and develop in the framework of matrices Fejer's theory for Fourier series.

Abstract. Anon-negativetriangularmatrixoperatorisconsideredinweightedLebesguespacesof sequences. Under some additional conditions on the matrix, some new weight characterizations for discrete Hardy type inequalities with matrix operator are proved for the case 1 < q < p < ∞. Some further results are pointed out. Mathematics subject classification (2000): 26D10, 26D15.

The sharpness of some recent integral inequalities is discussed and the corresponding extremal functions are pointed out. It is also proved that the cases of equality can equivalently be obtained by solving an integral equation of Volterra type with discontinous kernel chi(A) (t). This integral equation of independent interest is solved for every measurable set A on (a, b), -infinity < a > b > infinity.

Let f be a non-negative function defined on R-+(n) which is monotone in each variable separately. If 1 < p < infinity, g greater than or equal to 0 and nu a product weight function, then equivalent expressions forsup integral(R+)(n) fg / (R(+)(n)f(p)nu)(1/p)are given, where the supremum is taken over all such functions f.Variants of such duality results involving sequences are also given. Applications involving weight characterizations for which operators defined on such functions (sequences) are bounded in weighted Lebesgue (sequence) spaces are also pointed out.

A sharpened version of Carleman's inequality is proved. This result unifies and generalizes some recent results of this type. Also the “ordinary” sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman-type inequality with a more general measure is proved and this result may also be seen as a generalization of a continuous variant of Carleman's inequality, which is usually referred to as Knopp's inequality. A new elementary proof of (Carleman–)Knopp's inequality and a new inequality of Hardy–Knopp type is pointed out.

We characterize the inequality(integral (R+N) (fqu))(1/q) less than or equal to C(integral (R+N) (fpv))(1/p), 0 < q, p < infinityfor monotone functions f greater than or equal to 0 and nonnegative weights u and v. The case q < p is new and the case 0 < p < q < infinity is extended to a modular inequality with N-functions. A remarkable fact concerning the calculation of C is pointed out.