Question 8.11: Let z0 be in the upper half of the z plane. Show that the bi...

We have

|w|=\left|e^{i \theta_{0}}\left(\frac{z-z_{0}}{z-\bar{z}_{0}}\right)\right|=\left|\frac{z-z_{0}}{z-\bar{z}_{0}}\right|

From Fig. 8-73, if z is in the upper half plane, \left|z-z_{0}\right| \leq\left|z-\bar{z}_{0}\right|, the equality holding if and only if z is on the x axis. Hence, |w| \leq 1, as required.

The transformation can also be derived directly (see Problem 8.61).