For $R>0$ , $D_R=\{ z\in \mathbb{C} | |z|< R \}$. Let, $f,g: D_R \rightarrow \mathbb{C}$ analytic functions such that never are 0 in $D_R$.

Question

For $R>0$ , $D_R=\{ z\in \mathbb{C} | |z|< R \}$. Let, $f,g: D_R \rightarrow \mathbb{C}$ analytic functions such that never are 0 in $D_R$. Show that:

If for all $z\in \mathbb{C}$, $|f(z)|=|g(z)|$ then, there exists $\lambda \in \mathbb{C}$ with $|\lambda|=1$ and $f=\lambda g$.

Since f, g are analytic in $D_R$ then, satisfy the Cauchy-Riemann equations. If $f=u+iv$ and $g=p+iq$ then:

$u^2+v^2=p^2+q^2$

but I'm not sure how to proceed

Answer 1

We have that $\frac{f}{g}$ is analytic in $D_R$ and has constant norm $1$. But an analytic function with constant norm is constant.