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.
