1.  Our objective here is to deduce Liouville’s theorem from Parseval’s identity:

Let  be an entire function which is bounded by M.

a)  Applying Parseval’s identity , show that

b)  Verify that the left-hand side of the inequality in a) can have no term  with .

c)  Deduce that .

2.  Our objective here is to deduce Cauchy’s Inequality from Parseval’s identity:

Without loss of generality, let  be analytic in a disk  and let  be the circle. Suppose  in  for some positive constant .

As in problem 1. a) 

a)  Deduce that  for every integer .

b)  Now deduce

c) Finally deduce Cauchy’s Inequality.

3.  Our objective here is to deduce the Maximum Principle from Parseval’s identity:

Suppose  is an analytic function in , where the function  has a maximum M in .  We need to show that  is constant.

a)  Let  be such that.  Then there exists some r with  such that .  Applying Parseval’s identity, show that

b)  Deduce that  cannot have any terms except .  Thus for integers .  Consequently, .

4.  Our objective here is to give another proof of Liouville’s theorem by using Cauchy’s formula:

Let  be an entire function which is bounded by M.  Let .

a) Show that is an entire function.

b) Show that, for we have .

c)  Let w be a point and suppose R is large enough so that .   Using Cauchy’s formula, show that .

d) Deduce that .