Does there exist a function $f(z)$ holomorphic in $\mathbb{C}\backslash\{0\}$, such that
$$\left|f(z)\right|\geq\frac{1}{\sqrt{\left|z\right|}}$$
for all $z\in\mathbb{C}\backslash\{0\}?$
I'm not really sure on how to proceed or which particular theorems I should look at.