If $f:\mathbb{C} \rightarrow \mathbb{C}$ is a differentiable function on $|f(z)|\ge 1$ everywhere, what can one conclude about $f$?
Does $|f(z)|\ge 1$ mean the magnitude of $f(z)$ is greater than 1? If so, what can I conclude about $f$?
If $f:\mathbb{C} \rightarrow \mathbb{C}$ is a differentiable function on $|f(z)|\ge 1$ everywhere, what can one conclude about $f$?
Does $|f(z)|\ge 1$ mean the magnitude of $f(z)$ is greater than 1? If so, what can I conclude about $f$?
One can conclude that it is constant. Since $|f(z)|\geq 1$ for all $z$, then $1/f(z)$ is defined for all $z$, and is also analytic. Moreover, $|1/f(z)|\leq 1$. The conclusion then follows by Liouville's Theorem.
There is a result that follows immediately from liouville theorem(so essentially this solution does not differ from the above,but maybe makes just more instantaneous the understanding that the function is constant):the image of an entire noncostant function is a dense subset of $\mathbb{C}$,thus your function is constant.