Prove $|f^{\prime}(a)|< \frac{Im f(a)}{Im a} $ for analytic self mapping on upper half plane

Question

$f:H \to H$ is a analytic mapping, where $H$ is the upper half plane, then $a \in H$, prove the inequality : $|f^{\prime}(a)|< \frac{Im f(a)}{Im a} $. I tried to use Schwarz Lemma to solve it, but I could not get the $Im f(a)$ part, I have no other ideas. Thanks in advance for any help

Answer 1

Here is a solution. I assume you know about Cayley's map.