Let $1<p<2$. I'm trying to prove the inequality

$$ |a+b|^q+|a-b|^q\leq 2\big( |a|^p + |b|^p \big)^{q-1} $$ where $\frac{1}{p}+\frac{1}{q}=1$.

Following this paper, I am able to prove the inequality for real $a,b$.

I'm missing the step to complex.

I have tried to set $x=|a+b|^2$ and $y=|a-b|^2$ (and other combinations) and then use the real inequality for $x,y$ with no success.

This is not exactly a duplicate of On the second Clarkson's inequality because that one asked from a "from scratch" proof while I'm only asking for one step.

  • $\begingroup$ I found this answer which makes the work. One can found a proof of the required Hölder inequalities here (in French). If you are interested, I wrote a fully detailed proof in giulietta in the french part (search for "Clarkson"). $\endgroup$ – Laurent Claessens May 7 at 20:10

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