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Part CCSIR NET December 2025schwarz-pick-gives-a-specific-sharp-bound-not-the-tightest-looking-number-among-the-choices

Schwarz pick gives a specific sharp bound not the tightest looking number among the choices

Let 𝔻={z∈Cz\in\mathbb{C}:|z|<1} and f:𝔻→𝔻 be a holomorphic function which satisfies f(−1/2)=0. Which of the following statements are necessarily true?

  1. A.|f(−1/5)| ≤ 1/5
  2. B.|f(−1/5)| ≤ 1/3✓
  3. C.|f′(−1/2)| ≤ 1/2
  4. D.|f′(−1/2)| ≤ 4/3✓

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: ℂ and the unit disc are biholomorphic (both are simply connected)

More on this topic

The chapter behind this: Conformal maps and Möbius transformations — free to read

From Zeros and Mappings › Conformal maps, Möbius transformations, Schwarz lemma

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