Let $\Delta$ denote the Laplacian. I am trying to prove that if $f=u+iv$ is holomorphic on an open set $U\subset \mathbb{C}$ and $f$ is nonvanishing, then $$\Delta (\vert f\vert^p)=p^2\vert f\vert^{p-2}\Bigg\vert \frac{\partial f}{\partial z} \Bigg\vert^2, \text{any}\,p>0.\,\,\,\,\,(*)$$ This is Exercise 44 of Ch.1 in Function Theory of one Complex Variable, by Greene & Krantz.
Using the fact that $f$ is holomorphic (i.e., that $\partial u/\partial x=\partial v/\partial y$ & $\partial v/\partial x = -\partial u/\partial y)$ , I can rewrite the right-hand side of $(*)$ as $$p^2\vert f\vert^{p-2}\Bigg\vert \frac{1}{2}(\partial u/\partial x+\partial v/\partial y)+\frac{i}{2}(\partial v/\partial x-\partial u/\partial y) \Bigg\vert^2= p^2\vert f\vert^{p-2}\Bigg\vert \partial u/\partial x-i\partial u/\partial y \Bigg\vert^2=\,\,\,\,\,\,\,\,\,p^2\vert f\vert^{p-2}\Big((\partial u/\partial x)^2+(\partial v/\partial y)\Big)^2$$
However, I am not clear on how to proceed with the left-hand side: here's what I have $\Delta(\vert f \vert ^p)=\Delta(\vert u+iv\vert^p)=\frac{\partial^2}{\partial x^2}\vert u+iv\vert^p+\frac{\partial^2}{\partial y^2}\vert u+iv\vert^p=\frac{\partial}{\partial x}\Big[p(u+iv)/\vert u+iv\vert)*\frac{\partial}{\partial x}\vert u+iv \vert^{p-1}\Big] +\frac{\partial}{\partial y}\Big[p(u+iv)/\vert u+iv\vert)*\frac{\partial}{\partial y}\vert u+iv \vert^{p-1} \Big]$
Can someone give a hint on how to proceed with the left hand side? My only idea is to use brute force and to continue to differentiate, but then I'd end up doing that indefinitely.
One way to do this without "brute force" is to write $\Delta = 4 \partial_z \partial_{\bar{z}}$, and use the fact that since $f$ is holomorphic, then $\partial_z \bar{f} = 0$ and $\partial_{\bar{z}} f = 0$.
Further hint: $|f|^2 = f \bar{f}$.
Edit: Misread your work.