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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?
The hypothesis are contradictory.
Such an $f$ is entire.
There is a unique $f$ satisfying the given conditions.
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.