Let $f(z)$, and $g(z)$ be two analytic functions defined on a region $D\subset\mathbb{C}$. Suppose there exist a constant $r>0$ such that $|f(z)|^2+|g(z)|^2=r$ for every $z$ in $D$. Show that both functions must be constant.
How do I solve this problem? I just need some hints to solve this.
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Assume wlog $\overline{D(0,1)}\subset D$, just to simplify the notation. Say $f(z)=\sum a_n z^n$ and $g=\sum b_n z^n$.
Then $$|a_0|^2+|b_0|^2=r$$and $$\sum|a_n|^2+\sum |b_n|^2=\frac1{2\pi}\int_0^{2\pi}(|f(e^{it})|^2+|g(e^{it})|^2)\,dt=r,$$ so $a_n=b_n=0$ for $n>0$, hence $f$ and $g$ are constant (in the unit disk, and hence constant in $D$).
If you apply the Wirtinger derivative $\frac{\partial}{\partial \overline{z}}$ on both sides of $r=|f|^2+|g|^2=f\overline{f}+g\overline{g}$ you get $$f\overline{f'}+g\overline{g'}=0$$
where you need to take into account the product rule and Cauchy-Riemann equations: $\frac{\partial}{\partial \overline{z}}f=\frac{\partial}{\partial \overline{z}}g=0$
Therefore, $-\frac{f}{g}=\frac{\overline{g'}}{\overline{f'}}$
The left hand side is analytic, where $g\neq0$, and the right hand side is anti-analytic, when $f'\neq0$. Therefore, at all those points both of those quotients are constant.
If $f$ is a constant multiple of $g$, then $|f|^2$ is constant.