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 ∈ than the approach proposed by James A. Clarkson in 1936.

Statement of the inequalities

Let f, g ∈ L^p, where E is any measure space. If p ∈, then
The substitutions F = f + g and G = f − g yield the second of Hanner's inequalities:
For p ∈ 2, +∞) the inequalities are [reversed.
Note that for the inequalities become equalities which are both the parallelogram rule.