Prove that $ f(z) = 0 $ for all $ z \in \overline{D}(0,1) $.

Let $f$ be a continuous function on the closed unit disk and holomorphic inside it, such that $f(z) = 0$ for all $ z = \exp (it) $ with $0 ≤ t ≤ \pi/2 $. Prove that $ f(z) = 0 $ for all $ z \in \overline{D}(0,1) $.

I have worked this exercise for absurdity, but it has not reached any conclusion. I would like to know a more sophisticated test, perhaps by a direct method.

Solution 1:

$f$ is zero “only” on one quarter of the unit circle. The trick is to consider the function $$ g(z) = f(z)f(iz)f(-z)f(-iz) $$ which is zero on all of the unit circle, so that you can apply the maximum modulus function to $g$.

If you are familiar with the Schwarz reflection principle then you can also argue that $f$ can be continued analytically beyond the arc $\{e^{it} \mid 0 \le t \le \pi/2 \}$ and then apply the identity theorem.