Let h be a holomorphic function on Csetminus0 such that limztoinftyh(z)=0. For every nge1, consider fn(z)=sumk=1^nh(z^k). Let D denote the open unit disc zinC:z<1. Which of the following statements is necessarily true?

(D) The sequence fnnge1 converges pointwise to a holomorphic function on zinC:z1.

Q35Section B (Part B)35 of 120

Complex analysis — convergence of a series of rescaled functions

Let $h$ be a holomorphic function on $\mathbb{C}\setminus\{0\}$ such that $\lim_{|z|\to\infty}h(z)=0$ . For every $n\ge1$ , consider $f_n(z)=\sum_{k=1}^{n}h(z^k)$ . Let $\mathbb{D}$ denote the open unit disc $\{z\in\mathbb{C}:|z|<1\}$ . Which of the following statements is necessarily true?

Pick an option to check your answer.

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