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 In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. Definition A uniformly convex space is a no ...

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 A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...

. If ''p'' ∈ , 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'' ∈ , +∞) the inequalities are reversed (they remain non-strict). Note that for

p = 2

the inequalities become equalities which are both the parallelogram rule.

References

* * {{Functional analysis Banach spaces Inequalities Measure theory