Maximum is attained on the boundary Let $f_1, ... , f_n$ be holomorphic on a bounded domain $\Omega$, continuous on $\overline{\Omega}$.  Let $g = |f_1| + \cdots + |f_n|$.  The problem I couldn't get is to show that the maximum of $g$ is attained on the boundary of $\Omega$.  This should be some obvious application of the fact that the maximum of each $f_i$ occurs on the boundary, but I'm just not seeing it.
 A: Edit: ajotatxe helpfully pointed out my original answer is fundamentally flawed. I'm not sure what the preferred way to correct my work is, but here is my alternative answer:
Assume (for contradiction) that we found interior $z_i\in\Omega$ for which $g(z_b) < g(z_i)$ for all $z_b$ on the boundary of $\Omega.\  $ Then we have (complex) unit constants, $u_1,\ldots,u_n,$ ($|u_i|=1$) for which:
$$
g(z_i) = u_1f_1(z_i)+\ldots+u_nf_n(z_i)
$$
Now we notice two things; $g_i := u_1f_1+\ldots+u_nf_n$ is holomorphic on $\Omega$ / continuous on the boundary; $|g_i| \le g$ on the boundary of $\Omega$ because the $u_i$ are units and the triangle inequality applies to the absolute value.
So we are given the fact that the linear combination of $f_i$ has a maximum, $z_m$, on the boundary of $\Omega$ at which $|g_i(z_i)| \le |g_i(z_m)| \le g(z_m).\  $ But $|g_i(z_i)| = g(z_i)$ contradicts are assumption that $g(z_m) < g(z_i)$ and so the maximum of $g$ is attained on the boundary.
$$$$
Original, incorrect answer:
With the given holomorphic $f_1,\ldots,f_n$ we can construct a large  collection of holomorphic functions on $\Omega$:
$$ 
\begin{array}{llr} g_1=&+f_1+\ldots&+f_n \\
g_2=&-f_1+f_2+\ldots&+f_n \\
g_3=&+f_1-f_2+f_3+\ldots&+f_n \\
g_4=&-f_1-f_2+f_3+\ldots&+f_n \\
& \cdots & \\
g_{2^N}=&-f_1-f_2-\ldots&-f_{n-1}-f_n \end{array} 
$$
conclusion:

 Now we notice that $g(z) = \max\{ g_1(z),\ldots,g_{2^N}(z) \}$ and that the maximum of each $g_i$ occurs on the boundary of $\Omega.\  $ So it would  contradict at least one of the $g_i$ for $g$ to not also have its max on the boundary.

(There should be a nicer way to use abstractions regarding norms and $|f_i|$, but when I'm stuck with absolute values, I find replacing each $|y|\in\pm y$ to work on carefully managed domains)
