The maximum of $|f|+|g|$ occurs on the boundary

Let \(f\) and \(g\) be analytic on a domain \(D\) whose closure is compact in \(\mathbb C\). Show that \[ |f(z)|+|g(z)| \] takes its maximum on the boundary \(\partial D\).

Hint: consider \(f(z)e^{i\alpha}+g(z)e^{i\beta}\) for suitable \(\alpha,\beta\).