Hanner's inequalities explained

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

	p
\\|f+g\\|
	p

	p
\\|f-g\\|
	p

\geq(\|f\|_p+\|g\|_p)^p+|\|f\|_p-\|g\|_p|^p.

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

2^p(

	p
\\|F\\|
	p

	p
\\|G\\|
	p

)\geq(\|F+G\|_p+\|F-G\|_p)^p+|\|F+G\|_p-\|F-G\|_p|^p.

For p ∈ [2, +∞) the inequalities are reversed (they remain non-strict). Note that for <math>p = 2</math> the inequalities become equalities which are both the [[parallelogram rule]].

References

Clarkson. James A.. Uniformly convex spaces. Trans. Amer. Math. Soc.. 40. 1936. 3. 396 - 414. 10.2307/1989630. American Mathematical Society. 1989630. free.
Hanner. Olof. On the uniform convexity of L^p and ℓ^p. Ark. Mat.. 3. 3. 1956. 239 - 244. 10.1007/BF02589410. free.