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Aequationes mathematicae

, Volume 79, Issue 1–2, pp 157–172 | Cite as

Some new refined Hardy type inequalities with general kernels and measures

  • Shoshana AbramovichEmail author
  • Kristina Krulić
  • Josip Pečarić
  • Lars-Erik Persson
Article

Abstract

We state and prove some new refined Hardy type inequalities using the notation of superquadratic and subquadratic functions with an integral operator A k defined by
$$ A_kf(x):=\frac{1}{K(x)} \int\limits_{\Omega_2} k(x,y)f(y)d\mu_2(y), $$
where \({k: \Omega_1 \times \Omega_2 \to \mathbb{R}}\) is a general nonnegative kernel, (Ω1, μ 1) and (Ω2, μ 2) are measure spaces and
$$ K(x):=\int\limits_{\Omega_2} k(x,y)d\mu_2(y), \, x \in \Omega_1. $$
The relations to other results of this type are discussed and, in particular, some new integral identities of independent interest are obtained.

Mathematics Subject Classification (2000)

Primary 26D10 Secondary 26D15 

Keywords

Inequalities Hardy’s inequality Hardy–Hilbert’s inequality kernels measures Hardy type operators superquadratic function subquadratic function integral identities 

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Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  • Shoshana Abramovich
    • 1
    Email author
  • Kristina Krulić
    • 2
  • Josip Pečarić
    • 2
  • Lars-Erik Persson
    • 3
  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.Faculty of Textile TechnologyUniversity of ZagrebZagrebCroatia
  3. 3.Department of MathematicsLuleå University of TechnologyLuleåSweden

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