Hanner's inequalities
# 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.

