$f,g$ holomorphic functions and $|\,f(z)|^2+|\,f(z)|=|g(z)|^2+|g(z)|$

Question

If $f$, $g$ are holomorphic in a certain region of the complex plane and $$|\,f(z)|^2+|\,f(z)|=|g(z)|^2+|g(z)|$$ find the simplest possible relation between $f$ and $g$.

I tried to differentiate both sides, but i did not get any result.

Answer 1

Observe that $$ \lvert f(z)\rvert^2+\lvert f(z)\rvert=\lvert g(z)\rvert^2+\lvert g(z)\rvert $$ implies that $$ \Big(\lvert f(z)\rvert+\frac{1}{2}\Big)^2=\lvert f(z)\rvert^2+\lvert f(z)\rvert+\frac{1}{4}=\lvert g(z)\rvert^2+\lvert g(z)\rvert+\frac{1}{4}=\Big(\lvert g(z)\rvert+\frac{1}{2}\Big)^2. $$ Hence $$ \lvert f(z)\rvert+\frac{1}{2}=\lvert g(z)\rvert+\frac{1}{2} $$ and thus $ \lvert f(z)\rvert=\lvert g(z)\rvert. $

Claim. $\,g(z)=af(z)$, for some constant $\lvert a\rvert=1$.

Let $U$ be the domain of $f$ and $g$ which is a region in $\mathbb C$.

Case A. If $\,f\equiv 0$, then clearly, $g\equiv 0$, and the Claim holds.

Case B. If $\,f\not\equiv 0$, then $Z_f=\{z\in U:\,f(z)=0 \}$ does not have a limit point in $U$ and $U\smallsetminus Z_f$ is also a region in $\mathbb C$. In particular, for every $z\in U\smallsetminus Z_f$ $$ \lvert\, f(z)\rvert=\lvert g(z)\rvert, $$ and setting $h=g/f$, which is holomorphic in $U\smallsetminus Z_f$, we have that $$ \lvert\, h(z)\rvert=1, \,\,\,\text{for every $z\in U\smallsetminus Z_f$.} $$ Hence $h$ is constant (maximum principle) and thus $h(z)=a$, for some $\lvert a\rvert=1$. Consequently $$ g(z)=a\,f(z), \,\,\,\text{for every $z\in U\smallsetminus Z_f$,} $$ and this clearly holds in $Z_f$, since both $f$ and $g$ vanish there.