Let $f,g$ be holomorphic defined on a domain $U \subset \mathbb{C}$. Set
$$\varphi(z)=\vert f(z) \vert + \vert g(z) \vert.$$
If $\varphi$ attains its maximum value on $U$, then prove $f,g$ are constant functions.
So my idea is starting with the point $z_0 \in U$ such that
$$\bigg \vert \vert f(z) \vert + \vert g(z) \vert \bigg \vert \leq \bigg \vert \vert f(z_0) \vert + \vert g(z_0) \vert \bigg \vert \leq \vert f(z_0)\vert + \vert g(z_0) \vert, \space \text{for all $z \in U$}$$ but then I get stuck, is it possible for me to say since $\varphi$ is bounded, then by the maximum modulus principle then its constant? Or so I state $\varphi$ is constant since it is analytic and attains a max value? then argue that a constant which is the sum of two functions need be constant?
