Let $D^*=\{z \in \mathbb{C} : 0< |z| < 1\}$ and $f: D^* \to \mathbb{C}$ be an analytic function such that $|f(z)| \leq \frac{1}{|z|}$ on $D^*$ then such a $f$,
(a) necessarily extends to an analytic function on the unit disc
(b) necessarily has an essential singularity at $z=0$
(c) has at most a simple pole at $z=0$
(d) can have a pole of order $2$ at $z=0.$
I know basic definitions and concepts of analytic function and singularities but don't know how to approach such a problem, any hint is much appreciable
Thank you.