Is there a holomorphic nonzero function on a closed bounded connected subset of $\mathbb{C}$ which has infinitely many zeros on the boundary and at most a finitely many zeros in the interior (open connected subset of the closed set)?
If there are infinitely many zeros in the interior, I am not certain whether the function reduces identically to zero by using limit point compactness and constructing a sequence of points with an accumulation point which is certainly in the closure but may not be in the interior.
For clarity: By "holomorphic on a closed bounded connected subset", I mean holomorphic in the interior and continuous on the boundary since holomorphicity is defined only over open sets. For e.g., consider a nonzero function continuous on the closed unit disc and holomorphic on the open unit disc. Can it have only finitely many zeros on the open disc and infinitely many zeros on the boundary?
Thanks.