Analytic function on the unit semicircle that sends reals to reals and the arc to the arc must be rational. > Given a function $f$ that is analytic in the unit semicircle such that $f$ is real and continuous on the diameter of the semicircle, and such that $|f(z)|=1$ on the arc of the semicircle, prove that $f$ is a rational function. This is an exercises from an old Complex Analysis Qualifying Exam at my university.

Reflection in the diameter gives an analytic function $f_1$ on the unit disk with $\lvert f_1(z)\rvert = 1$ for all $z$ with $\lvert z\rvert = 1$. Then reflection in the unit circle -

$$f_2(z) = \begin{cases} f_1(z) &, \lvert z\rvert \leqslant 1 \\\ \dfrac{1}{\overline{f_1(1/\overline{z})}} &, \lvert z\rvert > 1\end{cases}$$

gives an entire _meromorphic_ function $f_2$. Since $f_2$ is either analytic at $\infty$ (if $f_1(0) \
eq 0$) or has a pole at $\infty$ (if $f_1(0) = 0$), it follows that $f_2$ is actually meromorphic on the sphere $\mathbb{C} \cup \\{\infty\\}$, hence rational.