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.
then the coefficients satisfy
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 (and that is the reason for the minus sign in the definition of ). 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
(Since the rearrangement of the terms is justified.) 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 (as is injective), 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.
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.