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 < +∞,
For 1 < p < 2,
i.e., q = p ⁄ (p − 1).
- Clarkson, James A. (1936), "Uniformly convex spaces", Transactions of the American Mathematical Society, 40 (3): 396–414, doi:10.2307/1989630, MR 1501880.
- Hanner, Olof (1956), "On the uniform convexity of Lp and ℓp", Arkiv för Matematik, 3 (3): 239–244, doi:10.1007/BF02589410, MR 0077087.
- Friedrichs, K. O. (1970), "On Clarkson's inequalities", Communications on Pure and Applied Mathematics, 23: 603–607, doi:10.1002/cpa.3160230405, MR 0264372.