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.
If p ∈ [1, 2], then 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).
the inequalities become equalities which are both the parallelogram rule.