I want to show that the inequality
$$2^{1-p}|x-y |^p \leq \left|\, x \vert x \vert^{p-1} - y \vert y \vert^{p-1} \,\right|$$
holds for every $x,y \in \mathbb{R}$ and every $p \geq 1$. I found this in my analysis paper but sadly I could not prove it. I tried to use the convexity of the function $x \mapsto \vert x \vert^p$ and also tried to use an integral representation. Can someone give me a hint or a link where this is shown? Thank you very much in advance.
There are basically two cases:
$x>0>y$, in which case replace $y$ by $-y$ and rewrite the inequality as $$\dfrac{x^p+y^p}{2} \geq \left (\dfrac{x+y}{2} \right )^p,$$ where $x,y>0$. This follows from the fact that the graph of $f(x)=x^p$ is concave up.
$x>y>0$. In this case rewrite the inequality as $$\dfrac{x^p-y^p}{2} \geq \left (\dfrac{x-y}{2} \right )^p,$$ but a stronger inequality holds in this case: $$x^p-y^p \geq (x-y)^p \geq 2(x-y)^p/2^p.$$