$f$ is analytic in $\Omega-{\{z_o}\}$, where $\Omega$ is a domain in $\mathbb C$. If $\text{Im}f(z)>= -B$ for all $z\in \mathbb C$. I am trying to figure out nature of singularity $f$ have at ${z_0}$.
Obviously, Casorati-Weirestrass comes in to play. We can conclude it can not have essential singularity at ${z_0}$. I can not see $f$ bounded near $z_0$ and neither I see $f(z)$ going to $\infty$ as $z\rightarrow z_0$. I am missing something here. Please help.