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 A. Clarkson in 1936.

Statement of the inequalities

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

$\|f+g\|_{p}^{p}+\|f-g\|_{p}^{p}\geq {\big (}\|f\|_{p}+\|g\|_{p}{\big )}^{p}+{\big |}\|f\|_{p}-\|g\|_{p}{\big |}^{p}.$

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

$2^{p}{\big (}\|F\|_{p}^{p}+\|G\|_{p}^{p}{\big )}\geq {\big (}\|F+G\|_{p}+\|F-G\|_{p}{\big )}^{p}+{\big |}\|F+G\|_{p}-\|F-G\|_{p}{\big |}^{p}.$

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

Note that for $p=2$ the inequalities become equalities which are both the parallelogram rule.

References

• Clarkson, James A. (1936). "Uniformly convex spaces". Trans. Amer. Math. Soc. 40 (3). American Mathematical Society: 396–414. doi:10.2307/1989630. JSTOR 1989630. MR1501880
• Hanner, Olof (1956). "On the uniform convexity of L^p and ℓ^p". Ark. Mat. 3 (3): 239–244. doi:10.1007/BF02589410. MR0077087