Hanner's inequalities
In mathematics, Hanner's inequalities are results in the theory of Lp spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of Lp spaces for p ∈ than the approach proposed by James A. Clarkson in 1936.Statement of the inequalities
Let f, g ∈ Lp, 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.
