Exercise 12.  Show that  .  

Solution 12.

See text and/or instructor's solution manual.

    For   and , we know by Theorem 1.3 that  ,  so all we need show is that  .  

    For  ,  let  .  Then  .  

This implies  ,  so that  .  

Therefore,  .

    The proof that     is similar.  

We are done.   

    Alternatively, we can give a proof based on Exercise 6 and Exercise10.  Noting that   is a positive constant, we have

        

This solution is complements of the authors.

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