Clarkson's inequalities

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, &mu;) be a measure space; let f, g : X → R be measurable functions in Lp. Then, for 2 ≤ p < +∞,


 * $$\left\| \frac{f + g}{2} \right\|_{L^p}^p + \left\| \frac{f - g}{2} \right\|_{L^p}^p \le \frac{1}{2} \left( \| f \|_{L^p}^p + \| g \|_{L^p}^p \right).$$

For 1 < p < 2,


 * $$\left\| \frac{f + g}{2} \right\|_{L^p}^q + \left\| \frac{f - g}{2} \right\|_{L^p}^q \le \left( \frac{1}{2} \| f \|_{L^p}^p +\frac{1}{2} \| g \|_{L^p}^p \right)^\frac{q}{p},$$

where


 * $$\frac1{p} + \frac1{q} = 1,$$

i.e., q = p ⁄ (p &minus; 1).

The case p ≥ 2 is somewhat easier to prove, being a simple application of the triangle inequality and the convexity of


 * $$x \mapsto x^p. $$