Example 2.16.  Show that if  ,  then  ,  where    is any complex number.

Solution.  As f merely reflects points about the y axis, we suspect that any -disk about the point would contain the image of the punctured -disk about    if  .  To confirm this conjecture, we let be any positive number and set  .  Then we suppose that  ,  which means that  .  The modulus of a conjugate is the same as the modulus of the number itself, so the last inequality implies that  .  This is the same as  .  Since   and  ,  this is the same as   ,  which in turn is the same as  , which is what we needed to show.

Explore Solution 2.16.

(c) 2006 John H. Mathews, Russell W. Howell