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Hanner's inequalities

In mathematics, Hanner's inequalities are results in the theory of L<sup>p</sup> spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of L<sup>p</sup> spaces for p&nbsp;∈&nbsp;(1,&nbsp;+∞) than the approach proposed by James A. Clarkson in 1936.

Statement of the inequalities

Let f,&nbsp;g&nbsp;∈&nbsp;L<sup>p</sup>(E), where E is any measure space. If p&nbsp;∈&nbsp;[1,&nbsp;2], then

The substitutions F&nbsp;=&nbsp;f&nbsp;+&nbsp;g and G&nbsp;=&nbsp;f&nbsp;&minus;&nbsp;g yield the second of Hanner's inequalities:

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

Note that for the inequalities become equalities which are both the parallelogram rule.

References