I'm trying to prove the following result in complex analysis:
If $f $ is an analytic and bijective function from the unit disc to an open connected region $A$ then the distance from $f(0)$ to the the border of $A$ is at least $|f'(0)|$.
The theorems I think I have to use are Schwarz Lemma and the Maximum Modulus Principle, but I can't find a good way to put it all together for this problem.
Unless I am mistaken, the distance from $f(0)$ to the border of $A$ is at most $|f'(0)|$:
Assume that $B_d(f(0)) \subset A$. Then $$ h(z) = f^{-1}(f(0) + dz) $$ maps the unit disk $\Bbb D$ into itself with $h(0) = 0$, and $$ h'(z) = \frac{d}{f'(f^{-1}(f(0) + dz)} \, . $$ Applying the Schwarz Lemma to $h$ gives $$ 1 \ge |h'(0)| = \frac{d}{|f'(0)|} $$ and therefore $d \le |f'(0)|$.
A lower bound of the distance from $f(0)$ to the boundary of $A$ is $\frac 14 |f'(0)|$ according to the Koebe quarter theorem.