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Another Practice Exam question:

Suppose that $f$ is a holomorphic function on the open disk of radius 5 centered at $0$, and suppose that $f$ maps the closed annulus $\{f(z) : 1 \leq |z|\leq 2\}$ into the open unit disk. Prove that the restriction of $f$ to $D (0;2) = \{ f(z) : |z| < 2\}$ has exactly one fixed point.

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  • $\begingroup$ What are your thoughts? $\endgroup$ – Clayton May 4 '13 at 0:27
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Hint: Rouché's theorem (Consider the zeros of $f(z)-z$).

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