The maximum of $|f|+|g|$ is in the boundary

I think $G$ is supposed to be bounded and hence $\overline{G}$ is compact. By continuity of $|f|+|g|$ and compactness of $\overline{G}$, there exists $z_0\in\overline{G}$ and $M\ge 0$, such that $|f|+|g|$ attains its maximum $M$ at $z_0$. Let $a,b\in\mathbb{C}$ with $|a|=|b|=1$, such that $|f(z_0)|=af(z_0)$ and $|g(z_0)|=bg(z_0)$. Define $h=af+bg$. Then $h$ is holomorphic on $G$ and continuous on $\overline{G}$. Moreover, $|h(z)|\le M$ on $\overline{G}$ and $h(z_0)=M$. Then the conclusion follows from maximum modulus principle.

If $\omega \in \mathbb{C}$ and $|\omega| = 1$ then $f + \omega g$ is holomorphic on $G$ and $|f + \omega g| \leq |f| + |g|$. Given some fixed $z \in G$ there is such an omega (depending on $z$) such that $|f(z) + \omega g(z)| = |f(z)| + |g(z)|$. Now apply the maximum modulus principle to $f + \omega g$.