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The ρ-quasiconcave Functions and Weighted Inequalities

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Inequalities and Applications

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 157))

Abstract

We present some facts from a general theory of ρ-quasiconcave functions defined on the interval I = (a, b) ⊆ ℝ and show how to use them to characterize the validity of weighted inequalities involving ρ-quasiconcave operators.

The research was supported by grant no. 201/05/2033 of the Grant Agency of the Czech Republic, by the INTAS grant no. 05-1000008-8157 and by the Institutional Research Plan no. AV0Z10190503 of the Academy of Sciences of the Czech Republic.

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References

  1. K. Andersen, Weighted inequalities for the Stieltjes transform and Hilbert’s double series, Proc. Royal. Soc. Edinburgh Sect. A 86 (1980), 75–84.

    Article  MathSciNet  Google Scholar 

  2. Yu.A. Brudnyi and N.Ya. Krugljak, Interpolation Functors and Interpolation Spaces1, North-Holland, Amsterdam, 1991.

    MATH  Google Scholar 

  3. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976.

    Book  Google Scholar 

  4. W.D. Evans, A. Gogatishvili and B. Opic, The reverse Hardy inequality with measures, Math. Inequal. Appl. 11 (2008), 43–74.

    MathSciNet  MATH  Google Scholar 

  5. W.D. Evans, A. Gogatishvili and B. Opic, Weighted inequalities involving ρ-quasiconcave operators, Preprint, Prague, 2003.

    Google Scholar 

  6. A. Gogatishvili, B. Opic and L. Pick, Weighted inequalities for Hardy-type operators involving suprema Collect. Math. 57 (2006), 227–255.

    MathSciNet  MATH  Google Scholar 

  7. A. Gogatishvili and L. Pick, Discretization and anti-discretization of rearrangement invariant norms, Publ. Mat. 47 (2003), 311–358.

    Article  MathSciNet  Google Scholar 

  8. M.L. Gol’dman, On integral inequalities on a cone of functions with monotonicity properties, Soviet Math. Dokl. 44 (1992), 581–587.

    MathSciNet  Google Scholar 

  9. M.L. Gol’dman, H.P. Heinig and V.D. Stepanov, On the principle of duality in Lorentz spaces, Can. J. Math. 48 (1996), 959–979.

    Article  MathSciNet  Google Scholar 

  10. S. Janson, Minimal and maximal method of interpolation, J. Funct. Anal. 44 (1981), 50–73.

    Article  MathSciNet  Google Scholar 

  11. P. Nilsson, Interpolation of Banach lattices, Studia Math. 82 (1985), 135–154.

    Article  MathSciNet  Google Scholar 

  12. B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes in Math. Ser. 219, Longman Sci. & Tech., Harlow (1990).

    MATH  Google Scholar 

  13. L.E. Persson, Relations between summability of functions and their Fourier series, Acta Math. Acad. Sci. Hung. 27 (1976), 267–280.

    Article  MathSciNet  Google Scholar 

  14. G. Sinnamon, A note on the Stieltjes transform, Proc. Royal. Soc. Edinburgh Sect. A 110 (1988), 73–78.

    Article  MathSciNet  Google Scholar 

  15. G. Sinnamon, Operators on Lebesgue spaces with general measures, Doctoral thesis, McMaster University, 1987.

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK

    William Desmond Evans

  2. Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Praha 1, Czech Republic

    Amiran Gogatishvili & Bohumír Opic

  3. Department of Mathematics and Didactics of Mathematics Pedagogical Faculty, Technical University of Liberec, Hálkova 6, 46117, Liberec, Czech Republic

    Bohumír Opic

Authors
  1. William Desmond Evans
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  2. Amiran Gogatishvili
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  3. Bohumír Opic
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Editor information

Editors and Affiliations

  1. Institut Mathematik, Universität Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Catherine Bandle

  2. Department Economic Analysis & Information Technology for Business, University of Debrecen, Kassai út.26, 4028, Debrecen, Hungary

    László Losonczi

  3. Department of Analysis Institute Mathematics & Informatics, University of Debrecen, Pf. 12, 4010, Debrecen, Hungary

    Attila Gilányi  & Zsolt Páles  & 

  4. Institut für Analysis, Universität Karlsruhe, Kaiserstrasse 12, 76128, Karlsruhe, Germany

    Michael Plum

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© 2008 Birkhäuser Verlag Basel/Switzerland

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Evans, W.D., Gogatishvili, A., Opic, B. (2008). The ρ-quasiconcave Functions and Weighted Inequalities. In: Bandle, C., Losonczi, L., Gilányi, A., Páles, Z., Plum, M. (eds) Inequalities and Applications. International Series of Numerical Mathematics, vol 157. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-8773-0_12

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