Aleksandra Čižmešija

On strengthened Hardy and Pólya-Knopp's inequalities

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 Polya-Knopps inequalities deriving them as special cases of the obtained general relations. We discuss Polya-Knopps inequality, compare it with Levin-Cochran-Lees inequalities and point out that these results are mutually equivalent. Finally, we also point out a reversed Polya-Knopp type inequality.

MULTIDIMENSIONAL HARDY-TYPE INEQUALITIES VIA CONVEXITY

Let an almost everywhere positive function Φ be convex for p >1 and p p ∈(0,1), and such that Ax p ≤Φ( x )≤ Bx p holds on for some positive constants A ≤ B . In this paper we derive a class of general integral multidimensional Hardy-type inequalities with power weights, whose left-hand sides involve instead of , while the corresponding right-hand sides remain as in the classical Hardy’s inequality and have explicit constants in front of integrals. We also prove the related dual inequalities. The relations obtained are new even for the one-dimensional case and they unify and extend several inequalities of Hardy type known in the literature.

On strengthened weighted Carleman's inequality

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 Carlemans inequality.

Classical Hardy’s and Carleman’s Inequalities and Mixed Means

The aim of this paper is to present an alternative approach to the classical discrete and integral Hardy’s and Carleman’s inequalities, considering their natural connection with discrete and integral power means and giving opportunity for their various generalizations. Following that idea, mixed means corresponding to chosen power means are introduced and relations between two different mixed means of the same type are established. A complete survey of recently proven mixed-means inequalities is given, accompanied with the basic ideas used in their proofs, and it is shown how can these relations be applied as a technique for deriving Hardy’s and Carleman’s inequalities. Further, two multivariable generalizations of reviewed one-dimensional integral results are given, one of them to balls and the other to cells in R n . Moreover, the best possible constants for all obtained inequalities are discussed.

On Bicheng-Debnath's generalizations of Hardy's integral inequality

We consider Hardys integral inequality and we obtain some new generalizations of Bicheng-Debnaths recent results. We derive two distinguished classes of inequalities covering all admissible choices of parameter k from Hardys original relation. Moreover, we prove the constant factors involved in the right-hand sides of some particular inequalities from both classes to be the best possible, that is, none of them can be replaced with a smaller constant.

Some new refinements of strengthened Hardy and Pólya–Knopp's inequalities

We prove a new general one-dimensional inequality for convex functions and Hardy–Littlewood averages. Furthermore, we apply this result to unify and refine the so-called Boass inequality and the strengthened inequalities of the Hardy–Knopp–type, deriving their new refinements as special cases of the obtained general relation. In particular, we get new refinements of strengthened versions of the well-known Hardy and Polya–Knopps inequalities.

On a new class of Hardy-type inequalities

In this paper, we generalize a Hardy-type inequality to the class of arbitrary non-negative functions bounded from below and above with a convex function multiplied with positive real constants. This enables us to obtain new generalizations of the classical integral Hardy, Hardy-Hilbert, Hardy-Littlewood-Pólya, and Pólya-Knopp inequalities as well as of Godunova’s and of some recently obtained inequalities in multidimensional settings. Finally, we apply a similar idea to functions bounded from below and above with a superquadratic function.MSC: 26D10, 26D15.

Some New Refined General Boas-Type Inequalities

We state and prove a new refined Boas-type inequality in a setting with a topological space and general 𝜎-finite and finite Borel measures. As a consequence of the result obtained, we derive a new class of Hardy- and Polya-Knopp-type inequalities related to balls in ℝ𝑛 and prove that constant factors involved in their right-hand sides are the best possible.

INEQUALITIES OF THE HILBERT TYPE IN

In this paper we state and prove a new general Hilbert-type inequality in

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