Classical Hardy’s and Carleman’s Inequalities and Mixed Means

  • Aleksandra Čižmešija
  • Josip Pečarić
Part of the Mathematics and Its Applications book series (MAIA, volume 517)


The aim of this paper is to present an alternative approach to the classical discrete and integral Hardy’s and Carleman’s inequalities, considering their natural connection with discrete and integral power means and giving opportunity for their various generalizations. Following that idea, mixed means corresponding to chosen power means are introduced and relations between two different mixed means of the same type are established. A complete survey of recently proven mixed-means inequalities is given, accompanied with the basic ideas used in their proofs, and it is shown how can these relations be applied as a technique for deriving Hardy’s and Carleman’s inequalities. Further, two multivariable generalizations of reviewed one-dimensional integral results are given, one of them to balls and the other to cells in R n . Moreover, the best possible constants for all obtained inequalities are discussed.


Positive Real Number Integral Power Classical Inequality Hardy Type Inequality Positive Measurable Function 
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Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Aleksandra Čižmešija
    • 1
  • Josip Pečarić
    • 2
  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia
  2. 2.Faculty of Textile TechnologyUniversity of ZagrebZagrebCroatia

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