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.
is analytic and injective in the punctured open unit disk
and has the power series representation then the coefficients
satisfy The idea of the proof is to look at the area uncovered by the image of
is a simple closed curve in the plane.
denote the unique bounded connected component of
The existence and uniqueness of
follows from Jordan's curve theorem.
is a domain in the plane whose boundary is a smooth simple closed curve
This follows easily, for example, from Green's theorem.
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
also equals to the average of the two expressions on the right hand side.
After simplification, this yields where
denotes complex conjugation.
and use the power series expansion for
the rearrangement of the terms is justified.)
Therefore, the right hand side is positive.
It only remains to justify the claim that
is positively oriented around
, we may write the expression for the winding number of
, and verify that it is equal to
is injective), the invariance of the winding number under homotopy in the complement of
implies that the winding number of
is positively oriented around
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.