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 A. Clarkson in 1936.
Statement of the inequalities Let f, g ∈ Lp(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 . {\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:
2 p ( ‖ F ‖ p p + ‖ G ‖ p p ) ≥ ( ‖ F + G ‖ p + ‖ F − G ‖ p ) p + | ‖ F + G ‖ p − ‖ F − G ‖ p | p . {\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 p = 2 {\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. MR 1501880 Hanner, Olof (1956). "On the uniform convexity of Lp and ℓp". Ark. Mat. 3 (3): 239–244. doi:10.1007/BF02589410. MR 0077087