Hanner's inequalities - Alchetron, The Free Social Encyclopedia

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

∥ f + g ∥ p p + ∥ f − g ∥ p p ≥ ( ∥ 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 ( ∥ F ∥ p p + ∥ G ∥ p p ) ≥ ( ∥ 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 p = 2 the inequalities become equalities, and the second yields the parallelogram rule.

References

Hanner's inequalities Wikipedia

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