My question stems from the following result about holomorphic functions on the unit disc:



> 
> "A function, continuous on the closed unit disc, holomorphic inside, and vanishing on an open subset of the boundary, vanishes identically. A simple proof of this well-known proposition is obtained by considering its Cauchy integral representation." (*Generalized Analytic Functions* by Richard Arens and I.M.Singer)
> 
> 
> 


There are other more direct proofs too, including the one by P.R. Chernoff, using a product of finitely many rotations of $f: \Bbb D \to \Bbb C$ and the identity theorem. See details [here](https://math.stackexchange.com/questions/443406/holomorphic-function-zeros-on-the-circle), for example.


In P.R. Chernoff's article, I found the statement of a stronger result - the set of zeros of $f: \Bbb D \to \Bbb C$ on $∂$ cannot have positive Lebesgue measure unless $f$ vanishes identically. The proof is more involved and can be found in Chapter 4 of *Banach Spaces Of Analytic Functions* by Kenneth Hoffman. See the Corollary on Pg. 52 of a Theorem on Pg. 51.


**Question:** It is natural to ask what happens if $\Bbb D$ is replaced by an arbitrary open (connected?) subset $\Omega \subset \Bbb C$. Does the result continue to hold, i.e., is it true that the set of zeros of a holomorphic function $f:\Omega\to \Bbb C$ on $\partial \Omega$ cannot have positive Lebesgue measure unless $f$ vanishes identically on $\Omega$? Does a weaker result hold? One may require $\Omega$ to be bounded.


Thanks a lot!



Boundary zeros of a holomorphic function $f: \Omega \to \Bbb C$
It is not clear in your question what "Lebesgue measure on $\partial\Omega$" really means.


Let us begin with the unit disk. In the unit disk, every bounded holomorphic function which is zero on a set $E$ of positive measure, in the sense that
$$\limsup\_{r\to 1}|f(re^{i\theta})|=0,\quad e^{i\theta}\in E,$$ vanishes. This is the direct consequence of Jensen's inequality
$$\log|f(0)|\leq\frac{1}{2\pi}\int\_{-\pi}^\pi\log|f(re^{i\theta})|\,d\theta, \quad r\in(0,1).$$
(If $f\neq 0$, you can find an automorphism $\phi$ of the unit disk
such that $g=f\circ\phi(0)\neq 0$. Applying Jensen's inequality to this $g$ and letting $r\to 1$ gives a contradiction.)


Now for other regions. If $\Omega$ is simply connected, you can use a conformal map of the unit disk onto $\Omega$. But then the question is what is the correct "measure" on $\partial \Omega$. It is called the harmonic measure. For a simply connected $\Omega$ it can be defined as the image of
the Lebesgue measure on the unit circle under the conformal map. But it can be defined for arbitrary region possessing
a Green function (in particular for any bounded region, or more generally for any region for which $\partial D$ contains a continuum). Let $G$ be the Green function, with
the pole at some point $z\_0\in\Omega$. Extend it to the whole plane by setting $G(z)=0,z\not\in\Omega$. The resulting function is subharmonic in ${\mathbf{C}}\backslash\{z\_0\}$,
and harmonic measure is the Riesz measure, $(2\pi)^{-1}\Delta G$ of this subharmonic function. Here $\Delta$ is in the sense of Schwarz distributions.


Now your question has the following general answer for arbitrary region possessing a Green function: if $f$ is bounded, and tends to zero as $z\to E$, where $E$ is a set
of positive harmonic measure then $f=0$.


The question remains how to characterize the sets of positive or zero harmonic measure in terms of some usual metric properties,
for example Hausdorff measures, in particular the $1$-dimensional measure when $\partial G$ is rectifiable etc.
The answer depends on the class of domains $\Omega$ considered,
and this is a subject of a very large body of literature,
of which I can recommend the modern book:


MR2150803 Garnett, John B.; Marshall, Donald E. Harmonic measure, Cambridge, 2005.


If $\Omega$ is simply connected, and $\partial\Omega$ is a rectifiable Jordan curve, then, according to a theorem of F. and M. Riesz, the sets of zero harmonic measure are the same as those of zero 1-Hausdorff measure.


F. and M. Riesz (1916), "Über die Randwerte einer analytischen Funktion", Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, pp. 27–44.


