Part CCSIR NET December 2024schwarz-supplies-the-geometric-majorant

Schwarz supplies the geometric majorant

Consider the disc 𝔻 = { $z \in \mathbb{C}$ : |z| < 1} and a non-constant holomorphic function f : 𝔻 → 𝔻. Suppose that f(0) = 0. For each n ≥ 1 and z ∈ 𝔻, define $f_{n}(z) = f(z^{n})$ . Which of the following statements are true?

A.The series $\sum(n\ge1) f_{n}$ converges only at z = 0.
B.The series $\sum(n\ge1) f_{n}$ converges pointwise only on a countable set E ⊆ 𝔻 but not on 𝔻 \ E.
C.The series $\sum(n\ge1) f_{n}$ converges pointwise at all points of 𝔻 but not uniformly on some compact subsets of 𝔻.
D.The series $\sum(n\ge1) f_{n}$ converges uniformly on all compact subsets of 𝔻 to a holomorphic function on 𝔻.✓ Correct

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, and why this one tests standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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Schwarz supplies the geometric majorant

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