I'd love to solve the following question:
Let $f$ be an analytic function defined in the domain $\{z | |Re (z)| < 1 , Im(z) > 0 \}$. If $$\lim_{\epsilon \to 0 , \epsilon >0} \sup_{1/3 <x< 2/3}|f(x+ i \epsilon)| = 0,$$ Then $f\equiv 0$.
By the assumption, we know that for each $x\in(1/3,2/3)$, there is a sequence $f(x+a_{x,n}i) \to 0$ as $n\to \infty$. To get the result, my guess is we need to use Maximum modulus principle. But I don't know how to start proof.