Let $\phi$ be a holomorphic function from the unit disk onto the upper half-plane such that $\phi(0)=\alpha$. Give a method to find an upper bound for $\lvert\phi ′(0)\rvert$?
To apply Schwarz's lemma, don't I just need to find a Möbius transformation that can be composed with $\phi$ so that the image of $\alpha$ under this Möbius transformation is $0$? (or am i wrong??)
Is there a general form for such Möbius transformations from the upper half-plane $H$ to the disc $D$ that sends $\alpha$ to $0$?