This is a question from an exam in an undergraduate complex analysis course.
We denote $G=\{z\in \mathbb{C}|Im(z)>0\}$ - the upper half plane. Given a holomorphic function $f\in Hol(G)$ such that $|f(z)|\leq e^{-1/|z|}$ for all $z\in G$, prove that $f=0$ identically on G.
The question provides a hint, which is to take a large positive $R>0$, and a sufficiently large natural number $N$, and define the function: $$g(z)=\prod_{k=0}^{N-1}f(e^{2\pi ik/N}*z+iR)$$ on the circle $D_R=\{z\in \mathbb{C}||z|<R\}$. I tried to define a sequence of such functions $g_N(z)$ and managed to show that they converge uniformly to 0 when $N$ approaches infinity through: $$0\leq |g_N(z)|=\prod_{k=0}^{N-1}|f(e^{2\pi ik/N}*z+iR)|\leq e^{-N/|z|}\rightarrow0$$However I got stuck afterwards, and couldn't find a way to apply the identitiy theorem as I wanted.