TITLE: If $|f|+|g|$ is constant on $D,$ prove that holomorphic functions $f,~g$ are constant on $D$.
QUESTION [6 upvotes]: Let $D\subseteq \mathbb{C}$ be open and connected and 
  $f,~g:D \rightarrow \mathbb{C}$ holomorphic functions such that 
  $|f|+|g|$ is constant on $D.$ Prove that $f,~g$ are constant on $D.$

Attempt. I noticed first that this problem is already set before, and is considered  to be duplicate (If $|f|+|g|$ is constant then each of $f, g$ is constant) and we are sent directly to problem sum of holomorphic functions. However, the last reference deals with the sum $|f|^2+|g|^2$, which I do not see how is connected to the sum $|f|+|g|$, as stated in our title (for example, the equality $|f|^2+|g|^2=(|f|+|g|)^2-2|f||g|$ is not helpful here).
Thanks in advance!

REPLY [2 votes]: Here is an answer that will help you to solve the problem the way that you already tried to do it.
First let's enumerate our equation
\begin{eqnarray}
\big|f\big| + \big|g\big| = c. \quad(1)
\end{eqnarray}
It's easy to see that f,g can't be zero on $D$, otherwise if for example there was $z_0\in D$ such as $f(z_0)= 0$ then you have that
$$\big|g(z_0)\big| = c$$
but you also have that
$$\big|g(z)\big|\leq c \quad \forall z\in D$$
so from maximum modulus principal you have that $g=c$ on $D$ so (1) gives us that $f = 0$ on $D$.
So we have that there exists some holomorphic functions  on $D$, $f_1,g_1$  such us
\begin{eqnarray}
&f_1^2=f \\
&g_1^2=g
\end{eqnarray}
(Theorem 13.11 page 274 from Real and Complex Analysis by Rudin).
Now you have what you were looking for, you have that
$$\big|f_1\big|^2+\big|g_1|^2 = c \quad (2)$$
and now all you need is to prove is that $f_1,g_1$ are constant.
Now you can continue as that link you mention, but there is one other way to do it.
We have that $f_1,g_1$ are holomorphic so they have a powerseries expansion
\begin{eqnarray}
&f_1(z) = \sum_{n=0}^{\infty}a_nz^n \\
&g_1(z) = \sum_{n=0}^{\infty}b_nz^n
\end{eqnarray}
so if we integrate on some circle $\gamma$ with centre  $0$ and radious $r$
(I suppose that $\;D = D(0,R)\;$ and $\;0<r<R\;$)
equation (2), Parseval's formula gives us that
\begin{eqnarray}
&\frac{1}{2\pi }\int_{\gamma}|f_1|^2dz + \frac{1}{2\pi }\int_{\gamma}|g_1|^2 dz =\frac{1}{2\pi}\int_{\gamma}cdz \Longrightarrow \\
&c_1 = \sum_{n=0}^{\infty}|a_n|^2r^{2n}+\sum_{n=0}^{\infty}|b_n|^2r^{2n} = \sum_{n=0}^{\infty}\Bigl(|a_n|^2+|b_n|^2\Bigr)r^{2n} \quad \forall r\in(0,R)
\end{eqnarray}
where $\; c_1\in \mathbb C\;$ is a constant, so all $a_n,b_n = 0$ for $n\geq 1$, and you have the result.