Exercise 6.  Prove that      for all  z.  Where does equality hold?  

Solution 6.

See text and/or instructor's solution manual.

Solution.   Set  .  We have    and  

                    ,

also
                    .  

Since    and for real numbers t,    is a non-negative increasing function, we have  .

Therefore,   .

We will have equality when       iff        iff    .

Therefore,     precisely when  ,  i.e., when z is a real number. 

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell