Let $L$ be the Laplace operator, i.e., $L(g)=g_{x x}+g_{y y}$. Prove that if $f: \Omega \rightarrow \mathbf{C}$ is analytic in a domain $\Omega$, then

$$L\left(|f(z)|^{2}\right)=4\left|f^{\prime}(z)\right|^{2}, \quad z \in \Omega .$$