Conformal Moduli and Annuli

Prove that every 1-1 conformal mapping of a disk onto another is given by a linear transformation.

Find all conformal transformations \(f(z)\) from the upper half disc \[ \{z\mid |z|<2 \text { and } \operatorname {Im}z>0\} \] onto the unit disc \(\{z\mid |z|<1\}\) such that \(f(i)=0\).

Let \(\Omega \subsetneq \mathbb C\) be a simply connected region, and fix \(z_0\in \Omega \). If \(\phi :\Omega \to D\) is a conformal map such that \(\phi (z_0)=0\), show that \[ |\phi '(z_0)| =\sup \{|f'(z_0)|:\ f:\Omega \to D\text { holomorphic}, f(z_0)=0\}. \]

Let \(H=\{z\in \mathbb C:\operatorname {Im}z>0\}\) be the upper half plane and let \(\Delta =\{z\in \mathbb C:|z|<1\}\) be the unit disk.

a. Find a conformal map from \(H\) to \(\Delta \) that takes \(i\) to \(0\).

b. Let \(f:H\to H\) be a holomorphic map with \(f(i)=i\). Show that \(|f'(i)|\le 1\) and that if \(|f'(i)|=1\) then \(f\) is of the form \(f(z)=\frac {az+b}{cz+d}\) with \(a,b,c,d\in \mathbb R\).

Assume that \(f\) is a one-to-one conformal mapping of the open disc \(|z|<1\) onto a regular \(n\)-gon with center at \(0\), and \(f(0)=0\). Prove that \(f(\rho z)=\rho f(z)\) where \(\rho \) is any \(n\)-th root of \(1\). Remember that conformal maps preserve angles.

Let \(U\) be the open unit disc in \(\mathbb C\). Let \(U^+\) be the intersection of \(U\) with the upper half plane.

(a) Let \(f(z)=\frac {1}{2}\left (z+\frac {1}{z}\right )\). Describe \(f(U^+)\).

(b) Give a conformal equivalence of \(U^+\) and \(U\).

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 \[ |f'(0)|\le 2|\Re f(0)|. \]

Find a conformal transformation of the disc \(\Delta =\{z:|z|<1\}\) onto itself, which maps \(0\) to \(-\frac {1}{2}\).