a. Let \(\Re \alpha <0\), and \(g(z)=\frac {\alpha -z}{\bar \alpha +z}\). Show that \(g\) is a conformal map of \(\{z\mid z\in \mathbb C\text { and }\Re z<0\}\) onto the open unit disk. (Hint: the ordered triple \(\{\alpha ,\frac {\alpha -\bar \alpha }{2},-\bar \alpha \}\) maps to \(\{0,1,\infty \}\) under \(g\).)
b. Let \(f\) be holomorphic on the closed unit disk and suppose that \(\Re f(z)<0\) for all \(z\) in the closed disk. Then show that \[ |f'(0)|\le 2|\Re f(0)|. \]