Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1133–1141 | Cite as

An Improvement of Hardy’s Inequality for Hilbert Space Operators

  • Mohsen Kian
  • Mohsen Rostamian DelavarEmail author


In this paper, using the notion of superquadratic functions and operator valued mappings, we give an improvement of the Hardy inequality for Hilbert space operators. Then we apply our results to the classical Hardy inequality. In particular, we obtain an estimation for the positive expression
$$\begin{aligned} \left( \frac{p}{p-1}\right) ^p\int _{0}^{\infty }f(x)^pdx - \int _{0}^{\infty }\left( \frac{1}{x}\int _{0}^{x}f(t)dt\right) ^pdx, \end{aligned}$$
where \(p\ge 1\) and f is a positive p-integrable function on \((0,\infty )\).


Hardy inequality p-integrable Weakly measurable mapping Superquadratic function 

Mathematics Subject Classification

Primary 26D15 Secondary 47A56 



The authors are thankful to the handling editor and the anonymous referees for useful comments and suggestions.


  1. 1.
    Abramovich, S., Jameson, G., Sinnamon, G.: Refining Jensen’s inequality. Bull. Math. Soc. Sci. Math. Roum. 47, 3–14 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Garayev, M.T., Gürdal, M., Saltan, S.: Hardy type inequaltiy for reproducing kernel Hilbert space operators and related problems. Positivity 21, 1615–1623 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Garayev, M.T., Gürdal, M., Okudan, A.: Hardy–Hilbert’s inequality and power inequalities for Berezin numbers of operators. Math. Inequal. Appl. 19, 883–891 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hansen, F.: Non-commutative Hardy inequalities. Bull. Lond. Math. Soc. 41(6), 1009–1016 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hardy, G.H.: Note on a theorem of Hilbert. Math. Z. 6, 314–317 (1920)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hardy, G.H.: Notes on some points in the integral calculus, LX. An inequality between integrals. Messenger Math. 54, 150–156 (1925)Google Scholar
  7. 7.
    Hardy, G., Littewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1967)Google Scholar
  8. 8.
    Kian, M.: Hardy–Hilbert type inequalities for Hilbert space operators. Ann. Funct. Anal. 3(2), 129–135 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kian, M.: On n Hardy operator inequality. Positivity 22, 773–781 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kian, M.: Operator Jensen inequality for superquadratic functions. Linear Algebra Appl. 456, 82–87 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kian, M., Dragomir, S.S.: Inequalities involving superquadratic functions and operators. Mediterr. J. Math. 11, 1205–1214 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Krnić, M., Pečarić, J., Perić, I., Vuković, P.: Recent Advances in Hilbert-Type Inequalities: A Unified Treatment of Hilbert-Type Inequalities. Element, Zagreb (2012)zbMATHGoogle Scholar
  13. 13.
    Kumar, S.: A Hardy-type inequality in two dimensions. Indag. Math. 20, 247–260 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pachpatte, B.G.: Mathematical Inequalities. Elsevier Science, Amsterdam (2005)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Basic SciencesUniversity of BojnordBojnordIran

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