Here are my problems:
Find an entire function $f(z)$ such that $|f(z)| < e^{\operatorname{Im} f(z)}$ for all $z \in \mathbb{C}$ and $f(0)=2$. I am trying to guess $2\cos(z), 2e^z$, etc, but they are all failed in the end.
Let $f(z)$ be analytic in the punctured disk $0<|z|<1$,and $f(1/n)=\sqrt{n}$ for every integer $n>1$. Prove that $f(z)$ has an essential singularity at $z=0$. I am trying to use the definition of essential singularity to prove. But I don't know how to express the $f(z)$
I am sorry that I wrote wrong condition for problem #1. The condition should be $|f(z)| > e^{\operatorname{Im} f(z)}$