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| In [[mathematics]], '''Clarkson's inequalities''', named after [[James A. Clarkson]], are results in the theory of [[Lp space|''L''<sup>''p''</sup> spaces]]. They give bounds for the ''L''<sup>''p''</sup>-[[norm (mathematics)|norm]]s of the sum and difference of two [[measurable function]]s in ''L''<sup>''p''</sup> in terms of the ''L''<sup>''p''</sup>-norms of those functions individually.
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| ==Statement of the inequalities==
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| Let (''X'', Σ, ''μ'') be a [[measure space]]; let ''f'', ''g'' : ''X'' → '''R''' be measurable functions in ''L''<sup>''p''</sup>. Then, for 2 ≤ ''p'' < +∞,
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| :<math>\left\| \frac{f + g}{2} \right\|_{L^p}^p + \left\| \frac{f - g}{2} \right\|_{L^p}^p \le \frac{1}{2} \left( \| f \|_{L^p}^p + \| g \|_{L^p}^p \right).</math>
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| For 1 < ''p'' < 2,
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| :<math>\left\| \frac{f + g}{2} \right\|_{L^p}^q + \left\| \frac{f - g}{2} \right\|_{L^p}^q \le \left( \frac{1}{2} \| f \|_{L^p}^p +\frac{1}{2} \| g \|_{L^p}^p \right)^\frac{q}{p},</math>
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| where
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| :<math>\frac1{p} + \frac1{q} = 1,</math>
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| i.e., ''q'' = ''p'' ⁄ (''p'' − 1).
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| The case ''p'' ≥ 2 is somewhat easier to prove, being a simple application of the [[triangle inequality]] and the [[convex function|convexity]] of
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| :<math>x \mapsto x^p. \, </math> | |
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| ==References==
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| *{{citation
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| | last = Clarkson | first = James A. | authorlink = James A. Clarkson
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| | doi = 10.2307/1989630
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| | issue = 3
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| | journal = [[Transactions of the American Mathematical Society]]
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| | mr = 1501880
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| | pages = 396–414
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| | title = Uniformly convex spaces
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| | volume = 40
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| | year = 1936}}.
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| *{{citation
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| | last = Hanner | first = Olof | authorlink = Olof Hanner
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| | doi = 10.1007/BF02589410
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| | issue = 3
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| | journal = [[Arkiv för Matematik]]
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| | mr = 0077087
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| | pages = 239–244
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| | title = On the uniform convexity of ''L''<sup>''p''</sup> and ''ℓ''<sup>''p''</sup>
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| | volume = 3
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| | year = 1956}}.
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| *{{citation
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| | last = Friedrichs | first = K. O. | authorlink = Kurt O. Friedrichs
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| | doi = 10.1002/cpa.3160230405
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| | journal = [[Communications on Pure and Applied Mathematics]]
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| | mr = 0264372
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| | pages = 603–607
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| | title = On Clarkson's inequalities
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| | volume = 23
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| | year = 1970}}.
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| ==External links==
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| * {{PlanetMath|urlname=ClarksonInequality|title=Clarkson inequality}}
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| [[Category:Banach spaces]]
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| [[Category:Inequalities]]
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