Maximum of sum of the moduli in a domain $D$ Assume $f_k(z), k=1,2,...,n$ to be $n$ holomorphic functions in a domain $D$. Is it possible $|f_1(z)| + |f_2(z)| + ... + |f_n(z)|$ takes strict local maximum within the domain $D$?
I know that if $n=1$, by maximum modulus principle, $|f_1(z)|$ cannot have a strict local maximum in $D$. But how to deal with the general case?
Thank you for any help!
 A: Suppose that $G(z)=|f_1(z)|+|f_2(z)|+\ldots+|f_n(z)|$ have a strict local maximum within $D$, say $z_0\in D$. That is, there is $r>0$ such that $D_{2r}(z_0)=\{z\in \mathbb{C} | \ \ |z-z_0|<2r \}\subset D$ and $G(z_0)>G(z)$ for all $z\in D_{2r} \backslash \{z_0\}$. For each $i=1,\ldots, n$, we can find $\theta_i\in [0,2\pi)$ such that 
$$
|f_i(z_0)| = e^{\theta_i} f_i(z_0).
$$ 
Consider $F(z)= e^{\theta_1} f_1(z) + e^{\theta_2} f_2(z) + \ldots + e^{\theta_n} f_n(z)$. Then $F$ is holomorphic. Hence, by Cauchy's integral formula, we have
$$
F(z_0)=\frac1{2\pi i} \int_{C_r(z_0)} \frac{F(w)}{w-z_0} dw
$$
where $C_r(z_0)$ is the circle or radius $r$ centered at $z_0$. 
By compactness of $C_r(z_0)$, we have 
$$
\max_{w\in C_r(z_0)} G(w) < G(z_0).
$$
Also, 
$$
G(z_0)=|F(z_0)|\leq \frac 1{2\pi} \int_{C_r(z_0)} \frac{G(w)}{|w-z_0|} |dw| <G(z_0). 
$$
This is a contradiction. Thus, $G(z)$ cannot have a strict local maximum within $D$. 
As I commented. This is essentially the proof of Maximum modulus principle. 
