Proving $\Delta(|f|^p)=p^2(|f|^{p-2}) |\frac{\partial{f}}{\partial{z}}|^2$ is true for any $p>0$ for a non-vanishing analytic function f in $\Omega$?

Question

This problem comes from "Function Theory of One Complex Variable" by Robert E. Greene and Stevan G. Kantz, p.26, exercise 44. It is a generalization of the problem before, which was answered here: Proving ∇2(|f|2)=4|∂f/∂z|2 via a "Laplacian Approach". I tried to follow the example from the above problem and extend it but I am fairly lost right now. Any explanations or partial hints would be welcome.