In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of Lp spaces. They give bounds for the Lp-norms of the sum and difference of two measurable functions in Lp in terms of the Lp-norms of those functions individually.
Statement of the inequalities Let (X, Σ, μ) be a measure space; let f, g : X → R be measurable functions in Lp. Then, for 2 ≤ p < +∞,
‖ f + g 2 ‖ L p p + ‖ f − g 2 ‖ L p p ≤ 1 2 ( ‖ f ‖ L p p + ‖ g ‖ L p p ) . {\displaystyle \left\|{\frac {f+g}{2}}\right\|_{L^{p}}^{p}+\left\|{\frac {f-g}{2}}\right\|_{L^{p}}^{p}\leq {\frac {1}{2}}\left(\|f\|_{L^{p}}^{p}+\|g\|_{L^{p}}^{p}\right).}
For 1 < p < 2,
‖ f + g 2 ‖ L p q + ‖ f − g 2 ‖ L p q ≤ ( 1 2 ‖ f ‖ L p p + 1 2 ‖ g ‖ L p p ) q p , {\displaystyle \left\|{\frac {f+g}{2}}\right\|_{L^{p}}^{q}+\left\|{\frac {f-g}{2}}\right\|_{L^{p}}^{q}\leq \left({\frac {1}{2}}\|f\|_{L^{p}}^{p}+{\frac {1}{2}}\|g\|_{L^{p}}^{p}\right)^{\frac {q}{p}},}
where
1 p + 1 q = 1 , {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1,}
i.e., q = p ⁄ (p − 1).
References Clarkson, James A. (1936), "Uniformly convex spaces", Transactions of the American Mathematical Society, 40 (3): 396–414, doi:10.2307/1989630, JSTOR 1989630, MR 1501880. Hanner, Olof (1956), "On the uniform convexity of Lp and ℓp", Arkiv för Matematik, 3 (3): 239–244, Bibcode:1956ArM.....3..239H, doi:10.1007/BF02589410, MR 0077087. Friedrichs, K. O. (1970), "On Clarkson's inequalities", Communications on Pure and Applied Mathematics, 23 (4): 603–607, doi:10.1002/cpa.3160230405, MR 0264372.
External links Clarkson inequality at PlanetMath.