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".