Show that there does not exist an analytic function $F : B(0,1) \to \mathbb{C}-\mathbb{Q} $ with $F(0)=i$ and $F(1/2)=-i$.
I have already used the open mapping theorem to show that if we assume towards a contrdiciton that such an $F$ exists then we must have $F : B(0,1) \to \mathbb{C}-\mathbb{R} $. I then thought of composing $F$ with the Mobius transformation
$M(z)=\frac{z-i}{z+i}$
that sends the real line to the unit circle and the half upper plane to the unit disc (without the boundary) and is such that $M(F(0))=0$ in an attempt to apply the Schwarz lemma. But I am stuck at this point. Tried to see if I could apply other theorems like the maximum principle but didn't have much progress.
I would prefer a hint or a nudge in the right direction rather than a complete answer. Thank you.