Showing a specific bound on the derivative of a holomorphic function at fixed point

Question

Let $f:D \to D$ be holomorphic where $D$ is the unit disk and such that $f(1/2)=1/2$. Show $|f'(1/2)| \le 3/4$.

I've used various forms of the Schwarz lemma and Schwarz-pick lemma via composing with the automorphism sending 1/2 to 0, but I'm still only able to prove that $|f'(1/2)| \le 1$ (which it itself is a corollary to Schwarz lemma in my text). Any suggestions?

Answer 1

The function $f(z) = z$ would appear to be a counterexample.