Uniqueness of a holomorphic function with certain boundary values on an arc

Question

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)

Answer 1

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.

Answer 2

I believe this is correct, but I may be wrong. Say we know the precise values of a holomorphic function everywhere in an arc and say the arc ranges from an angle of $\theta_0$ to $\theta_1$ as measured ( for example, but this is arbitrary) clock-wise from the origin. Then, we define the new function $f(\theta)$ as the values of the initial holomorphic function at an angle $\theta$ which is within the range from $\theta_0$ to $\theta_1$. Now, $f(\theta)$ can be interpreted as a real function defined within an interval. It must be analytic within this interval. From there, one can derive a power series of $f$ and extend it to all real values of $\theta$ ( values should cycle every $2\pi$ ). Also, from this power series, one can obtain the values of $f$ at complex numbers ( for now, do not over-think the idea of having complex angles). Let $g(z)$ be the initial holomorphic function. Then, by our initial definition, within the arc ( of radius, say, $r$) we have $$f(\theta)=g(re^{i\theta}) $$ and since we have $f$ for all complex $\theta$, we can choose a $\theta$ to make $re^{i\theta}$ whatever we want, and so we can obtain the function $g$. I hope I have explained it clearly. On a more general note, if one knows the behaviour of an analytic function in any connected subset of the complex plane, one can find the values of that function in the entire plane ( analytic continuations are, after all, unique). I hope that helped.

Answer 3

Suppose $f$ is analytic on the disc, whose limits at the boundary yields a set of zeros with non-empty interior, and $f$ is continuous on some neighborhood (in the closed disc) of one such interior accumulation point. By reflecting through the circular boundary we can extend $f$ to be analytic on a neighborhood (in the plane) of the accumulation point. As was indicated by a comment, this is essentially the Schwarz reflection principle, though in a slightly different form than usually stated. Thus the extension has an accumulation of zeros in its domain, and so is identically zero.