Complex inequality $||u|^{p-1}u - |v|^{p-1}v|\leq c_p |u-v|(|u|^{p-1}+|v|^{p-1})$ How does one show for complex numbers u and v, and for p>1 that
\begin{equation*}
||u|^{p-1}u - |v|^{p-1}v|\leq c_p |u-v|(|u|^{p-1}+|v|^{p-1}),
\end{equation*}
where $c_p$ is some constant dependent on p.  My intuition is to use some version of the mean value theorem with $F(u) = |u|^{p-1}u$, but I'm not sure how to make this work for complex-valued functions.  Plus there seems to be an issue with the fact that $F$ may not smooth near the origin.
For context, this shows up in Terry Tao's book Nonlinear Dispersive Equations: Local and Global Analysis on pg. 136, where it is stated without proof as an "elementary estimate".

Without loss, you can assume $|u|\le|v|$ and replace $u$ by $uv$ to reduce the problem to the situation $v=1$ and $|u|\le1$. Unless you need $c_p$ explicitly, then it's clear, yes?
