Hanner's inequalities

Template:Short description 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

$\Vert f+g{\Vert }_p^p+\Vert f-g{\Vert }_p^p\ge (\Vert f{\Vert }_p+\Vert g{\Vert }_p{)}^p+|\Vert f{\Vert }_p-\Vert g{\Vert }_p{|}^p.$

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

$2^p(\Vert F{\Vert }_p^p+\Vert G{\Vert }_p^p)\ge (\Vert F+G{\Vert }_p+\Vert F-G{\Vert }_p{)}^p+|\Vert F+G{\Vert }_p-\Vert F-G{\Vert }_p{|}^p.$

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

• Template:Cite journal Template:MathSciNet
• Template:Cite journal Template:MathSciNet