On Bicheng‐Debnath′s generalizations of Hardy′s integral inequality

Čižmešija, Aleksandra; Pečarić, Josip

doi:10.1155/s0161171201005816

Cited by 4 publications

(2 citation statements)

References 4 publications

(3 reference statements)

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“…in the monographs [13,22,23,25,26,27,28], expository papers [6,17,21], and the references cited therein. Besides, here we also emphasize the papers [2,4,5,7,8,9,18,19,24,29,32,33], all of which to some extent have guided us in the research we present here.…”

Section: Introductionmentioning

confidence: 77%

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

Čižmešija

Hussain

Pečarić

2009

Journal of Function Spaces and Applications

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Abstract. 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 Boas's 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 Pólya-Knopp's inequalities.

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Section: Introductionmentioning

confidence: 77%

“…They also obtained a related dual inequality, that is, an inequality with the outer integrals taken over b, ∞ and with the inner integral on the lefthand side taken over x, ∞ . These general inequalities provided an unified treatment of the strengthened Hardy and Pólya-Knopp's inequalities from [7,8] and [32,33].…”

Section: Introductionmentioning

confidence: 99%

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

Čižmešija

Hussain

Pečarić

2009

Journal of Function Spaces and Applications

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On strengthened Hardy and Pólya–Knopp's inequalities

Čižmešija

Pečarić

Persson

2003

Journal of Approximation Theory

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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 Po´lya-Knopp's inequalities deriving them as special cases of the obtained general relations. We discuss Po´lya-Knopp's inequality, compare it with Levin-CochranLee's inequalities and point out that these results are mutually equivalent. Finally, we also point out a reversed Po´lya-Knopp type inequality. r

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Some new refined Hardy-type inequalities with kernels

Čižmešija¹,

lreich²,

Pečarić³

2010

Journal of Mathematical Inequalities

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Abstract. By using the notion of the subdifferential of a convex function, we state and prove a new general refined weighted Hardy-type inequality for convex functions and the integral operator with a non-negative kernel. We point out that the obtained result generalizes and refines the classical one-dimensional Hardy's, Pólya-Knopp's, and Hardy-Hilbert's inequalities, as well as related dual inequalities. We show that our results may be seen as generalizations of some recent results related to Riemann-Liouville's and Weyl's operator, as well as a generalization and a refinement of the so-called Godunova's inequality.Mathematics subject classification (2010): 26D10, 26D15.

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On Bicheng‐Debnath′s generalizations of Hardy′s integral inequality

Cited by 4 publications

References 4 publications

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

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

On strengthened Hardy and Pólya–Knopp's inequalities

Some new refined Hardy-type inequalities with kernels

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