It is said that the following result is true using Schwarz lemma:
Suppose $g$ is a holomorphic map from the unit disc to the half plane $Re(z)\leq\beta$ such that $g(0)=0$ and $G$ is a conformal mapping from the unit disc onto the same half plane with $G(0)=0$ as well. Then $|g’(0)|\leq|G’(0)|.$
Can anyone tell me how this is true or give a reference for the same?
Apply the Schwarz Lemma to $f=G^{-1}\circ g$.