Reference for an inequality between $|x+y|^p$, $|x|^p$, $|y|^p$, and $|x-y|^p$

Question

I am interested in an inequality for real numbers $x,y$ and $1<p<\infty$ that it should say something like $$ |x+y|^p -|x|^p - |y|^p \leq (1-2^{1-p})|x-y|^p. $$

Is this inequality (or something similar) valid?

Where can I find a proof?