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?