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Encyclopedia > Hardy's inequality

Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if is a sequence of non-negative real numbers which is not identically zero, then for every real number p > 1 one has The feasible regions of linear programming are defined by a set of inequalities. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, the real numbers may be described informally in several different ways. ...

An integral version of Hardy's inequality states if f an integrable function with non-negative values, then In calculus, the integral of a function is an extension of the concept of a sum. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ...

The equality holds if and only if f(x) = 0 almost everywhere. In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...


Hardy's inequality was first published (without proof) in 1920 in a note by Hardy[1]. The original formulation was in an integral form slightly different than the above.


See also

  • Carleman's inequality

Notes

  1. ^ Hardy, G.H., Note on a Theorem of Hilbert, Math. Z. 6 (1920), 314-317.

References

  • Hardy, G. H.; Littlewood. J.E.; PĆ³lya, G. (1952). Inequalities, 2nd ed. Cambridge University Press. ISBN 0521358809. 
  • Kufner, Alois; Persson, Lars-Erik (2003). Weighted inequalities of Hardy type. World Scientific Publishing. ISBN 9812381953. 

 
 

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