Is it true that if a holomorphic function in the unit disk converges uniformly to the $0$ function some connected arc of the unit circle, this function is globally null?
If that is true, this would stand for a uniqueness fact and so how to recover some holomorphic function from its boundary value on an arc (if the uniform convergence still holds on the arc)
Best Answer
Let the (open) arc on which the boundary values of $f$ vanish be $A$. Since the boundary values of $f$ on $A$ are real, by the Schwarz reflection principle we know that the function
$$g(z) = \begin{cases} \;\,f(z) &, \lvert z\rvert < 1\\ \quad 0 &, z \in A\\ \overline{f(1/\overline{z})} &, \lvert z\rvert > 1\end{cases}$$
is holomorphic on the connected open set $\mathbb{D} \cup A \cup (\mathbb{C}\setminus \overline{\mathbb{D}})$. Since $g$ vanishes on a non-discrete set, the identity theorem yields $g \equiv 0$, in particular $f \equiv 0$ follows.