A New Approach to the Sawyer and Sinnamon Characterizations of Hardy's Inequality for Decreasing Functions

Abstract Some Hardy type inequalities for decreasing functions are characterized by one condition (Sinnamon), while others are described by two independent conditions (Sawyer). In this paper we make a new approach to deriving such results and prove a theorem, which covers both the Sinnamon result and the Sawyer result for the case where one weight is increasing. In all cases we point out that the characterizing condition is not unique and can even be chosen among some (infinite) scales of conditions.

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Hardy Inequalities on the Real Line

Canadian Mathematical Bulletin ◽

10.4153/cmb-2010-091-8 ◽

2011 ◽

Vol 54 (1) ◽

pp. 159-171 ◽

Cited By ~ 1

Author(s):

Mohammad Sababheh

Keyword(s):

Real Line ◽

Hardy’S Inequality ◽

Hardy Inequalities ◽

The Real ◽

Hardy Type Inequalities ◽

Hardy Type ◽

Hardy's Inequality ◽

Littlewood Conjecture ◽

The Relationship

AbstractWe prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

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On a new class of Hardy type inequalities

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500022095 ◽

1987 ◽

Vol 105 (1) ◽

pp. 265-274 ◽

Cited By ~ 6

Author(s):

B.G. Pachpatte

Keyword(s):

Integral Inequalities ◽

Hardy’S Inequality ◽

Hardy Type Inequalities ◽

New Class ◽

Hardy Type ◽

Hardy's Inequality

SynopsisIn this paper we establish a new class of integral inequalities which originate from the well-known Hardy's inequality. The analysis used in the proofs is quite elementary and is based on the idea used by Levinson to obtain generalisations of Hardy's inequality.

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Generalizations of Hardy Type Inequalities by Abel–Gontscharoff’s Interpolating Polynomial

Mathematics ◽

10.3390/math9151724 ◽

2021 ◽

Vol 9 (15) ◽

pp. 1724

Author(s):

Kristina Krulić Himmelreich ◽

Josip Pečarić ◽

Dora Pokaz ◽

Marjan Praljak

Keyword(s):

Convex Functions ◽

Upper Bounds ◽

Higher Order ◽

Hardy’S Inequality ◽

Hardy Type Inequalities ◽

Hardy Type ◽

Hardy's Inequality

In this paper, we extend Hardy’s type inequalities to convex functions of higher order. Upper bounds for the generalized Hardy’s inequality are given with some applications.

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A new approach to Hardy-type inequalities

Archiv der Mathematik ◽

10.1007/s00013-014-0722-5 ◽

2015 ◽

Vol 104 (2) ◽

pp. 165-176

Author(s):

Adam Osȩkowski

Keyword(s):

New Approach ◽

Hardy Type Inequalities ◽

Hardy Type

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Generalized sharp Hardy type and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.40.2009.604 ◽

2009 ◽

Vol 40 (4) ◽

pp. 401-413 ◽

Cited By ~ 2

Author(s):

Shihshu Walter Wei ◽

Ye Li

Keyword(s):

Fixed Point ◽

Euclidean Space ◽

Riemannian Manifolds ◽

Compact Support ◽

Hardy’S Inequality ◽

Sharp Constants ◽

Hardy Type ◽

Hardy's Inequality ◽

Nirenberg Inequality

We prove generalized Hardy's type inequalities with sharp constants and Caffarelli-Kohn-Nirenberg inequalities with sharp constants on Riemannian manifolds $M$. When the manifold is Euclidean space we recapture the sharp Caffarelli-Kohn-Nirenberg inequality. By using a double limiting argument, we obtain an inequality that implies a sharp Hardy's inequality, for functions with compact support on the manifold $M $ (that is, not necessarily on a punctured manifold $ M \backslash \{ x_0 \} $ where $x_0$ is a fixed point in $M$). Some topological and geometric applications are discussed.

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