# Clarkson inequality for complex numbers

Let $$1. 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.

• 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"). – Laurent Claessens May 7 at 20:10