On Hardy q -inequalities

Lech Maligranda; Ryskul Oinarov; Lars-Erik Persson

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 3, page 659-682
  • ISSN: 0011-4642

Abstract

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Some q -analysis variants of Hardy type inequalities of the form ∫ 0 b x α - 1 ∫ 0 x t - α f ( t ) d q t p d q x ≤ C ∫ 0 b f p ( t ) d q t with sharp constant C are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.

How to cite

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Maligranda, Lech, Oinarov, Ryskul, and Persson, Lars-Erik. "On Hardy $q$-inequalities." Czechoslovak Mathematical Journal 64.3 (2014): 659-682. <http://eudml.org/doc/262153>.

@article{Maligranda2014,
abstract = {Some $q$-analysis variants of Hardy type inequalities of the form \[ \int \_0^b \bigg (x^\{\alpha -1\} \int \_0^x t^\{-\alpha \} f(t) \{\rm d\}\_q t \bigg )^\{p\} \{\rm d\}\_q x \le C \int \_0^b f^p(t) \{\rm d\}\_q t \] with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.},
author = {Maligranda, Lech, Oinarov, Ryskul, Persson, Lars-Erik},
journal = {Czechoslovak Mathematical Journal},
keywords = {inequality; Hardy type inequality; Hardy operator; Riemann-Liouville operator; $q$-analysis; sharp constant; discrete Hardy type inequality; Hardy type inequality; Hardy operator; Riemann-Liouville operator; -analysis; sharp constant; discrete Hardy type inequality},
language = {eng},
number = {3},
pages = {659-682},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Hardy $q$-inequalities},
url = {http://eudml.org/doc/262153},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Maligranda, Lech
AU - Oinarov, Ryskul
AU - Persson, Lars-Erik
TI - On Hardy $q$-inequalities
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 659
EP - 682
AB - Some $q$-analysis variants of Hardy type inequalities of the form \[ \int _0^b \bigg (x^{\alpha -1} \int _0^x t^{-\alpha } f(t) {\rm d}_q t \bigg )^{p} {\rm d}_q x \le C \int _0^b f^p(t) {\rm d}_q t \] with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.
LA - eng
KW - inequality; Hardy type inequality; Hardy operator; Riemann-Liouville operator; $q$-analysis; sharp constant; discrete Hardy type inequality; Hardy type inequality; Hardy operator; Riemann-Liouville operator; -analysis; sharp constant; discrete Hardy type inequality
UR - http://eudml.org/doc/262153
ER -

References

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