Area theorem (conformal mapping)


In the mathematical theory of conformal mappings, the area theorem
gives an inequality satisfied by
the power series coefficients of certain conformal mappings.
The theorem is called by that name, not because of its implications, but rather because the proof uses
the notion of area.

Statement

Suppose that is analytic and injective in the punctured
open unit disk
and has the power series representation
then the coefficients satisfy

Proof

The idea of the proof is to look at the area uncovered by the image of.
Define for
Then is a simple closed curve in the plane.
Let denote the unique bounded connected component of
. The existence and
uniqueness of follows from Jordan's curve theorem.
If is a domain in the plane whose boundary
is a smooth simple closed curve,
then
provided that is positively oriented
around.
This follows easily, for example, from Green's theorem.
As we will soon see, is positively oriented around
. After applying the chain rule
and the formula for, the above expressions for
the area give
Therefore, the area of also equals to the average of the two expressions on the right
hand side. After simplification, this yields
where denotes complex conjugation. We set and use the power series
expansion for, to get
Now note that is if
and is zero otherwise. Therefore, we get
The area of is clearly positive. Therefore, the right hand side
is positive. Since, by letting, the
theorem now follows.
It only remains to justify the claim that is positively oriented
around. Let satisfy, and set
, say. For very small, we may write the
expression for the winding number of around,
and verify that it is equal to. Since, does
not pass through when
, the invariance
of the winding number under homotopy in the complement of
implies that the winding number of
around is also.
This implies that and that
is positively oriented around, as required.

Uses

The inequalities satisfied by power series coefficients of conformal
mappings were of considerable interest to mathematicians prior to
the solution of the Bieberbach conjecture. The area theorem
is a central tool in this context. Moreover, the area theorem is often
used in order to prove the Koebe 1/4 theorem, which is very
useful in the study of the geometry of conformal mappings.