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This is a variant of question Show that if f is analytic in $|z|\leq 1$, there must be some positive integer n such that $f(\frac{1}{n})\neq \frac{1}{n+1}$.

(i). Show that if $f$ is analytic in the unit disc $D=\{z\in C : |z|<1\}$, then there exists an integer $n$ such that $f(1/n)\neq1/(n+1)$
(ii) Does there exist an analytic function $g$ in the unit disc $D=\{z\in C : |z|<1\}$ such that $g(1/n)=g(-1/n)=1/n^2$ for all positive integers $n$?
(iii) Does there exist an analytic function $g$ in the unit disc $D=\{z\in C : |z|<1\}$ such that $h(1/n)=h(-1/n)=1/n^3$ for all positive integers $n$?

Thoughts thus far: For (i) the linked problem seems to be the most concise way to show this, but we are noy concerned with the point $z = -1$ as it was in the linked problem so the.same method cannot be used as that in the linked problem For (ii) and (iii), isn't the answer no because there is a singularity at the origin. However, perhaps I am missing some lemma not shown in class that shows an analytic function may have a removable discontinuity (but I doubt this is the case).

Thank you in advance for any help that you may provide.

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$z\mapsto \frac{z}{1+z}$ is analytic on $D=\{z\in\mathbb C\colon|z|<1\}$. –  Hagen von Eitzen Oct 14 '12 at 17:14

1 Answer 1

up vote 1 down vote accepted

More or less obvious candidate analytic functions for your conditions are

  • $f(z)=\frac z{1+z}$ with pole at $z=-1$, i.e. luckkily outside $D$.
  • $g(z)=z^2$

For the third, we must have $h(z)=z^3$ because that holds for the sequence $z_n=\frac1n$ with $z_n\to 0$. However, this contradicts $h(-\frac1n)=\frac1{n^3}$. Hence there is no such function.

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Thank you for your response. However, isn't $f(z)=\frac{z}{1+z}$ not helpful because we do not obtain a contradiction like the linked problem? –  ABC Bach Oct 14 '12 at 19:57
    
As your answer above suggests, part I is contradicted with a counter example. Thank you for.showing me how to come up candidate functions for these types of problems. –  ABC Bach Oct 15 '12 at 21:47

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