If $f,g: U \rightarrow \Omega$ are holomorphic, $f(0)=g(0)$ and $f$ is 1-1&onto, then $f$ has larger image of a disk than that of $g$.

Question

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?

Answer 1

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