Some Proofs of the Classical Integral Hardy Inequality (original) (raw)

On Some Integral Inequalities of Hardy-Type Operators

Advances in Pure Mathematics, 2013

In recent time, hardy integral inequalities have received attentions of many researchers. The aim of this paper is to obtain new integral inequalities of hardy-type which complement some recent results.

On weighted norm integral inequality of G. H. Hardy’s type

In this paper, we give a necessary and sufficient condition on Hardy's integral inequality: X [T f ] p wdµ ≤ C X f p vdµ ∀f ≥ 0 (1) where w, v are non-negative measurable functions on X, a non-negative function f defined on (0, ∞), K(x, y) is a non-negative and measurable on X ×X, (T f)(x) = ∞ 0 K(x, y)f (y)dy and C is a constant depending on K, p but independent of f. This work is a continuation of our recent result in [9].

On a Generalization of Hardy-Hilbert's Integral Inequality

2009

A generalization of Hardy-Hilbert's integral inequality was given by B.Yang in . The main purpose of the present article is to generalize the inequality. As applications, the reverse, the equivalent form of the inequality, some particular results and the generalization of Hardy-Littlewood inequalities are derived. Mathematics subject classification: 26D15.

Some new inequalities involving the Hardy operator

Mathematische Nachrichten, 2019

In this paper we derive some new inequalities involving the Hardy operator, using some estimates of the Jensen functional, continuous form generalization of the Bellman inequality and a Banach space variant of it. Some results are generalized to the case of Banach lattices on (0, ], 0 < ≤ ∞.

Some variants of the Hardy inequality

Applicable Analysis, 2020

We study an affine version of the Hardy inequality which is truly stronger than the usual Hardy inequality. We also set up a sharp version of the Hardy inequality under Sobolev-Lorentz norm.

Some new refined Hardy type integral inequalities

arXiv: Classical Analysis and ODEs, 2015

In this paper, by using Jensen's inequality and Chebyshev integral inequality, some generalizations and new refined Hardy type integral inequalities are obtained. In addition, the corresponding reverse relation are also proved.

Further integral inequalities related to Hardy's inequality

The paper gives generalisations of Hardy's integral inequality that further extend results of Mohapatra and Russell (Aequationes Math. 28 (1985), 199-207) and other authors by using α-submultiplicative or α-supermultiplicative functions.

Some new extensions of Hardy's inequality

International Journal of Nonlinear Analysis and Applications, 2014

In this study, by a non-negative homogeneous kernel k we prove some extensions of Hardy's inequalityin two and three dimensions

Generalization of Hardy-Hilbert's Inequality and Applications

Kyungpook mathematical journal, 2010

In this paper, by introducing some parameters we establish an extension of Hardy-Hilbert's integral inequality and the corresponding inequality for series. As an application, the reverses, some particular results and their equivalent forms are considered.

New equivalent conditions for Hardy-type inequalities

Mathematica Bohemica

We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties.

Generalized Hardy inequalities

Journal of Mathematical Analysis and Applications, 2003

We show that the generalized Hardy inequality k |b kf (n k )| C f H 1 holds for f ∈ H 1 and certain (b k ) ∈ l r (2 q ∞) whenever (n k ) ∞ k=1 ⊂ N satisfies appropriate growth conditions dependent on r.

On a Strengthened Form of a General Hardy-Type Inequality

Journal of Mathematical Sciences: Advances and Applications, 2020

In this paper, we employed the concept of superquadratic functions to obtain a class of Hardy-type inequalities involving a more general Hardy operator. Our results are new and in some cases refine and improve similar results in the literature.