Hanner's inequalities

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" ∈ [1, +∞) than the approach proposed by James Clarkson in 1936.

tatement of the inequalities

Let "f", "g" ∈ "L""p"("E"), where "E" is any measure space. If "p" ∈ [1, 2] , then

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

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

:2^p ig( |F|_p^p + |G|_p^p ig) geq ig( |F+G|_p + |F-G|_p ig)^p + ig| |F+G|_p-|F-G|_p ig|^p.

For "p" ∈ [2, +∞) the inequalities are reversed (they remain non-strict).

Note that for "p" = 2 the inequalities become equations, and the second yields the parallelogram rule.

References

* cite journal
last = Clarkson
first = James A.
title = Uniformly convex spaces
journal = Trans. Amer. Math. Soc.
volume = 40
year = 1936
issue = 3
pages = 396–414
issn = 0002-9947
doi = 10.2307/1989630 MathSciNet|id=1501880
* cite journal
last = Hanner
first = Olof
title = On the uniform convexity of "L""p" and "ℓ""p"
journal = Ark. Mat.
volume = 3
issue = 3
year = 1956
pages = 239–244
issn = 0004-2080
doi = 10.1007/BF02589410 MathSciNet|id=0077087


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