Hint for Q 3 p. 97 Ahlfors:

The problem asks you to conformally map the plane with a half-circle arc cut out, to the complement of the unit disk in the plane. In particular, viewed as maps from the Riemann sphere to itself, the point at infinity should be fixed.

Compare this problem to the conformal map (and its inverse) constructed in the bottom of page 94.  These maps show the conformal equivalence of the plane minus the [-1,1] segment (think of this as a slit plane) and the unit disk (with one more composition, one can replace the unit disk with the complement of the disk).  These maps relate to Example 7 in Stein-Shakarchi, which is the subject of another suggested exercise.  Understanding these maps is a preliminary problem to this exercise.

One can solve the problem with a composition of successive conformal maps just like in Exercise 8, Ch. 8 of Stein-Shakarchi.  Here is one succession of maps that works.