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 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

${\displaystyle \|f+g\|_{p}^{p}+\|f-g\|_{p}^{p}\geq {\big (}\|f\|_{p}+\|g\|_{p}{\big )}^{p}+{\big |}\|f\|_{p}-\|g\|_{p}{\big |}^{p}.}$

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

${\displaystyle 2^{p}{\big (}\|F\|_{p}^{p}+\|G\|_{p}^{p}{\big )}\geq {\big (}\|F+G\|_{p}+\|F-G\|_{p}{\big )}^{p}+{\big |}\|F+G\|_{p}-\|F-G\|_{p}{\big |}^{p}.}$

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

Note that for ${\displaystyle p=2}$ the inequalities become equalities which are both the parallelogram rule.

References

• Clarkson, James A. (1936). "Uniformly convex spaces". Trans. Amer. Math. Soc. 40 (3). American Mathematical Society: 396–414. doi:10.2307/1989630. JSTOR 1989630. MR1501880
• Hanner, Olof (1956). "On the uniform convexity of L^p and ℓ^p". Ark. Mat. 3 (3): 239–244. doi:10.1007/BF02589410. MR0077087

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Lp spaces

Basic concepts

• Banach & Hilbert spaces
• L^p spaces
• Measure
  • Lebesgue
• Measure space
• Measurable space/function
• Minkowski distance
• Sequence spaces

L¹ spaces

• Integrable function
• Lebesgue integration
• Taxicab geometry

L² spaces

• Bessel's
• Cauchy–Schwarz
• Euclidean distance
• Hilbert space
• Parseval's identity
• Polarization identity
• Pythagorean theorem
• Square-integrable function

${\displaystyle L^{\infty }}$ spaces

• Bounded function
• Chebyshev distance
• Infimum and supremum
  • Essential
• Uniform norm

Maps

• Almost everywhere
• Convergence almost everywhere
• Convergence in measure
• Function space
• Integral transform
• Locally integrable function
• Measurable function
• Symmetric decreasing rearrangement

Inequalities

• Babenko–Beckner
• Chebyshev's
• Clarkson's
• Hanner's
• Hausdorff–Young
• Hölder's
• Markov's
• Minkowski
• Young's convolution

Results

• Marcinkiewicz interpolation theorem
• Plancherel theorem
• Riemann–Lebesgue
• Riesz–Fischer theorem
• Riesz–Thorin theorem

For Lebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality

Applications & related

• Bochner space
• Fourier analysis
• Lorentz space
• Probability theory
• Quasinorm
• Real analysis
• Sobolev space
• *-algebra
  • C*-algebra
  • Von Neumann