Suppose $f: D=\{z \in \mathbb{C}: \vert z \vert <1 \} \rightarrow D$ is analytic and $f(0)=0$ and $f(1/2)=\frac{1}{2\sqrt2}+i\frac{1}{2\sqrt2}$. Find the value of $f(1/4)$?
I check $\vert f(1/2)\vert=1/2$,but how I can use this result to find the value of $f(1/4)$?
You apply the Schwarz lemma to deduce that $f(z)=az$ for some $a\in S^1$. Since $$f\left(\frac12\right)=\frac1{2\sqrt2}+\frac i{2\sqrt2},$$what must be the value of $a$? Then $f\left(\frac14\right)=\frac a4$.