I'm trying to work on the following application of the Schwarz Lemma, however, I can make no progress in finding the solution beyond finding a map that allows me to apply the Schwarz Lemma. Can Anyone help?
Suppose $f$ is analytic on the disk and makes the disk into the disk. Suppose also that $f(1/2) = 1/8$. What is the largest possible value for $|f(3/4i)|$?
In order to apply the Schwarz lemma, Ive set $$ g(x) = (\phi_{1/8} \circ f \circ \phi_{1/2})(z), $$ where $$ \phi_{\alpha}(z) = \frac{\alpha - z}{1 - \overline{\alpha} z}, |\alpha| < 1. $$
Then $g$ maps 0 to 0 and so we can apply the Schwarz Lemma to $g$ to see that $|g(z)| \leq |z|$ for all $|z| < 1$.
The issues is the composition of $f$ on the left by $\phi_{1/8}$; it prevents me to getting anything beyond $|g(3/4i)| \leq 3/4$.