Hanner's inequalities
Encyclopedia
In mathematics
, Hanner's inequalities are results in the theory of Lp spaces
. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity
of Lp spaces for p ∈ (1, +∞) than the approach proposed by James Clarkson in 1936.
The substitutions F = f + g and G = f − g yield the second of Hanner's inequalities:
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Hanner's inequalities are results in the theory of Lp spaces
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity
Uniformly convex space
In mathematics, uniformly convex spaces are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A...
of Lp spaces for p ∈ (1, +∞) than the approach proposed by James Clarkson in 1936.
Statement of the inequalities
Let f, g ∈ Lp(E), where E is any measure space. If p ∈ [1, 2], thenThe substitutions F = f + g and G = f − g yield the second of Hanner's inequalities:
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.