If $f_1(z), ···, f_n(z)$ are analytic in domain $D$, and continuous to its boundary. Prove that $ \varphi (z) = \lvert f_1(z) \rvert + ··· + \lvert f_n(z) \rvert$ is continuous and achieves its maximum value on the boundary.
The question emerges in the chapter "taylor series" from the book, but I have no clue.
