Question : Prove by use of Schwarz's lemma that every one-to-one conformal mapping of a disc onto another (or a half plane) is given by a linear fractional transformation.
I have known that there exists LFT such that it maps unit disc onto itself, but if holomorphic function $f$ is a 1-1 mapping of a disc onto another disc, can we conclude $f$ is LFT?
My try is simplifying the question as
Every one-to-one conformal mapping of a unit disc onto itself is given by a linear fractional transformation.
Am I right? Sincerely thanks for your help!