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Part CCSIR NET December 2025multiply-by-z-to-clear-the-pole-then-apply-maximum-modulus-to-the-resulting-entire-function

Multiply by z to clear the pole then apply maximum modulus to the resulting entire function

Let f be an entire function. Consider the function g given by g(z) = f(z) − 1/z for z∈Cz\in\mathbb{C}{0}. Which of the following statements are necessarily true?

  1. A.The function g has a pole at 0.✓
  2. B.If g(α)=0g(\alpha)=0, then |α\alpha|≠1.
  3. C.The function g has only finitely many zeros.
  4. D.max∣z∣=1max_{|z|=1} |g(z)| ≥ 1✓

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.

See pricing

50 are analysed free — try those first.

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

More on this topic

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

From Cauchy Theory › Liouville, Morera, maximum modulus principle

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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