I'm working on the following exercise (not homework) from Ahlfors' text:
" If $f(z)$ is analytic in $|z| \leq 1$ and satisfies $|f| = 1$ on $|z| = 1$, show
that $f(z)$ is rational."
I already know about the reflection principle for the case of a half plane, so I tried using the "Cayley transform" $$T (\zeta)=\frac{\zeta-i}{\zeta+i}$$
Which maps the closed upper half plane onto the closed unit disk with $1$ removed.
I defined $$g(\zeta)=(T^{-1} \circ f \circ T)(\zeta)=i\frac{1+f \left( \frac{\zeta-i}{\zeta+i} \right)}{1-f \left( \frac{\zeta-i}{\zeta+i} \right)},$$
And tried to apply the reflection principle in the book. $g$ is indeed analytic in the upper half plane, but for $\zeta \in \mathbb R$, I'm afraid that $g$ might get infinite (because on the boundary, $f$ takes values on the unit circle). If so, it will not be continuous and not even real, and the reflection principle is not applicable.
Am I missing something here? After all Ahlfors does mention in the text a generalized reflection principle for arbitrary circles $C,C'$.
Thanks
Best Answer
First note that the hypothesis implies that $f$ has only a finite number of zeros in the unit disk $\mathbb{D}$, say $\alpha_1, \dots, \alpha_n$. Consider now the function $$B(z):=\prod_{j=1}^n \frac{z-\alpha_j}{1-\overline{\alpha_j}z}.$$ This is a finite Blaschke product and $|B(z)| =1$ for all $z \in \partial \mathbb{D}$. Since $B$ has the same zeros as $f$, it follows that both $f/B$ and $B/f$ are analytic in $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$. By the maximum principle applied to both quotients, we deduce that $|f/B|=1$ everywhere in $\mathbb{D}$, so that $f/B$ is a unimodular constant $\lambda$, by the open mapping theorem. Therefore $f= \lambda B$, a rational function.