I am studying about automorphism group of simply connected surfaces. My reference book is "Dynamics in One Complex Variable" by John Milnor. In page 5, automorphism group of $\hat{\mathbb{C}}$ (Riemann sphere), called $\mathcal{G} ( \hat{\mathbb{C}} )$ is introduced.
There, for $g \in \mathcal{G} ( \hat{\mathbb{C}} )$ with $g(0) = 0$ and $g(\infty) = \infty$, he states that $\frac{g(z)}{z}$ is bounded!
But its boundedness is not clear for me. Could anyone help me?
