# which of the following is/are true for $f$? [duplicate]

Let $f$ be a meromorphic function on $\mathbb C$ such that $|f(z)|\ge|z|$ at each $z$ where $f$ is holomorphic. Then which of the following is/are true?

2. Such an $f$ is entire.

3. There is a unique $f$ satisfying the given conditions.

4. There is an $A\in\mathbb C$ with $|A|\ge1$ such that $f(z)=Az$ for each $z\in\mathbb C.$

I think 2 and 4 are correct. But I can't proceed rigorously. Neither do the discussion https://math.stackexchange.com/questions/631244/f-be-a-meromorphic-function-on-mathbbc-ni-fz-ge-z-forall-z is complete.

## marked as duplicate by Henrik, John B, Servaes, colormegone, Daniel W. FarlowJun 5 '16 at 0:37

Note that $f$ has no zeros, except possibly at the origin. Define $$g(z) = \frac z {f(z)}$$
Since $g$ is bounded near $0$ (even if $f$ has a pole at zero, $g$ is still bounded), we can extend $g$ to be holomorphic in a neighborhood of zero. Anywhere else $f$ has a pole, $g$ has a zero; thus, $g$ is holomorphic everywhere, or entire.
Finally, $g$ is bounded on $\mathbb{C}$, so must be constant. The correct choices then follow immediately.