Hanner's inequalities

In mathematics, Hanner's inequalities are results in the theory of L^p spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of L^p spaces for p ∈ (1, +∞) than the approach proposed by James Clarkson in 1936.

Statement of the inequalities

Let f, g ∈ L^p(E), where E is any measure space. If p ∈ [1, 2], then

$\|f%2Bg\|_p^p %2B \|f-g\|_p^p \geq \big( \|f\|_p %2B \|g\|_p \big)^p %2B \big| \|f\|_p-\|g\|_p \big|^p.$

The substitutions F = f + g and G = f − g yield the second of Hanner's inequalities:

$2^p \big( \|F\|_p^p %2B \|G\|_p^p \big) \geq \big( \|F%2BG\|_p %2B \|F-G\|_p \big)^p %2B \big| \|F%2BG\|_p-\|F-G\|_p \big|^p.$

For p ∈ [2, +∞) the inequalities are reversed (they remain non-strict).

Note that for p = 2 the inequalities become equalities, and the second yields the parallelogram rule.

References

Clarkson, James A. (1936). "Uniformly convex spaces". Trans. Amer. Math. Soc. (American Mathematical Society) 40 (3): 396–414. doi:10.2307/1989630. JSTOR 1989630. MR 1501880
Hanner, Olof (1956). "On the uniform convexity of L^p and ℓ^p". Ark. Mat. 3 (3): 239–244. doi:10.1007/BF02589410. MR 0077087