1.  CAUCHY’S INEQUALITY,  LIOUVILLE’S THEOREM, AND PARSEVAL’S IDENTITY

1.1          Cauchy’s Inequality

Theorem 1. (Cauchy’s Inequality). Let be analytic in a disk and let be the circle . Suppose in for some positive constant . Then

Proof. By the previous corollary we get

The result follows after letting  .

1.2          Liouville’s Theorem

Theorem 2. (Liouville’s Theorem).  A bounded entire function must be constant.

Proof. By the previous corollary for and for every we can write . Let Then . Since was arbitrary, But then and thus  is constant.

1.3          Parseval’s Identity

Theorem 3. (Parseval’s Identity).  Let and for let

.

Then for , we have .

Proof.  Consider the partial sum

of the series  .

Then

Multiplying out, integrating from 0 to , and using when and when j=k we get

. 

Since uniformly on @ and is bounded on , it follows that

uniformly on .  Letting gives Parseval’s identity.