For a $\alpha \in \mathbb{R}$, I am considering the function $f: \mathbb{R}^n \to \mathbb{R}^n$ defined as
$$ f_{\alpha}(x) = |x|^{\alpha} \operatorname{sign}(x) $$
where the absolute value, the sign, and the multiplication are pointwise.
For $p\in \mathbb{N}$ it is fairly easy to show that
$$ || f_p(x_1) - f_p(x_2)||_2 \leq p \; ||x_1 - x_2||_2 \; \max(||x_1||_2, ||x_2||_2)^{p-1}.$$
Question: Does the above inequality also hold for $\alpha = 1/p$ with $p\in \mathbb{N}$ and for a general constant $C>0$ on the rhs (instead of $p$)?
Remark 1: Numerical experiments suggest that this is true. However, my proof for the case $p \in \mathbb{N}$ only works for $\alpha \ge 1$ and I can show the general result only for $n=1$.
Remark 2: I would be perfectly happy to replace $\max(||x_1||_2, ||x_2||_2)^{\alpha-1}$ by $\max(1,||x_1||_2, ||x_2||_2)^{\alpha-1}$ if that helps.