Schwarz's Lemma

Question

Suppose that $f$ is analytic on the unit disk $D$ ,$f(0)=0$ and $f(D)⊂[-1,1]×[-0.01i,0.01i]$ .

Prove that $|f'(0)|<1$ .

In order to use Schwarz's lemma we have to map the rectangle onto the unit circle I'm stuck here!!

Any help please? Thanks.

Answer 1

In the question linked HERE, one proves that if $f$ analytic on the unit disc, $f(0)=0$, $|\Re f(z)| \le A$, then $|f'(0)| \le \frac{4A}{\pi}$ which is sharp.

Here using $g(z)=if(z)$, $g(0)=0, |\Re g(z)| \le \frac{1}{100}$ so $|f'(0)| =|g'(0)| \le \frac{1}{25 \pi} <1$

Note that mapping the unit circle onto a rectangle is given (via the upper plane) from the Schwarz Christoffel formulas, so it is not that easy to use those explicitly