Part CCSIR NET June 2024bounded-below-means-liouville-on-the-reciprocal

Bounded below means liouville on the reciprocal

Suppose that f is an entire function such that |f(z)| ≥ 2024 for all $z \in C$ . Which of the following statements are necessarily true?

A.f(z) = 2024 for all $z \in C$
B.f is a constant function✓
C.f is an injective function
D.f is a bijective function

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

50 are analysed free — try those first.

Related counterexample: A bounded holomorphic function on an unbounded domain is constant

More on this topic

identity theorem pigeonholeDecember 2023
liouville obstructionDecember 2023
a fractional growth rate rounds to an integer both timesDecember 2024
omitting a disc invites liouville on the reciprocalDecember 2024
vacuous hypothesisJune 2024
level sets are discrete but f may not vanishJune 2024

The chapter behind this: Liouville, Morera and the maximum principle — free to read

From Cauchy Theory › Liouville, Morera, maximum modulus principle

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