injectivity of holomorphic function based on its restriction

Asked Sep 13 '14 at 13:13

Active Sep 13 '14 at 13:13

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Let $f$ be holomorphic on $\mathbb D$ . Suppose there is an annulus {$z : 0 < r < |z| < 1$} in $\mathbb D$ such that $f$ restricted to this annulus is one-to-one.Show that $f$ is one-to-one on $\mathbb D$.

(Source: - Gamelin, Chapter 8)

N.B. :- I couldn't arrive at the solution using the hint given there. So, I am posting it here.

Thank You!

asked Sep 13 '14 at 13:13

user92360

Is $\mathbb{D}$ the unit disk ${ z \in \mathbb{C}:|z| <1}$? – Crostul Sep 13 '14 at 13:19
Yes !! @ Crostul. – user92360 Sep 13 '14 at 14:55
Possible duplicate of this question. – Crostul Sep 13 '14 at 17:37
Sorry...may be a possible duplicate.. but instead of something topological solution, I think Open Mapping Theorem or something like that is to be used, since this is a problem from the Complex Analysis book by Gamelin. I am looking for that solution using those theories. – user92360 Sep 13 '14 at 18:40

injectivity of holomorphic function based on its restriction

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