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I'm working on the RCA of rudin but having a difficulty in the following problem:

Suppose $f$ and $g$ are holomorphic mappings of $U$(the unit circle centered at 0) into $\Omega$, $f$is one to one and $f(U)= \Omega$, and $f(0)=g(0)$. Prove that

$g(D(0;r)) \subset f(D(0,r))$ for each $0 < r < 1 $.

I tried to use the fact that both images are open and espeially, $f(D(0,r))$ is a simply connected region, but have no idea to begin. Can anyone give me a hint?

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    Thanks joriki. It was m$y$ mistake that I did not obey the essential rules! I edited the title and tags. Please let me know if there is another mistake:)2012-12-12

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Hint: Apply Schwarz lemma to $f^{-1}\circ g$.

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    @richard Got it. Thank you!2013-03-02