In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 theory of conformal mappings, the area theorem
gives an inequality satisfied by
the power series coefficient

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s 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

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

.

Statement

Suppose that is analytic

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

 and injective

Injective function

In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...

 in the punctured
open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 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

Jordan curve theorem

In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...

.

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

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

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

Winding number

In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...

 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.

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

Koebe 1/4 theorem

In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states that the image of an injective analytic function f:\mathbb D\to\mathbb C from the unit disk \mathbb D onto a subset of the complex plane contains the disk whose center is f\, and whose radius is |f\,'|/4. The theorem is...

, which is very
useful in the study of the geometry of conformal mappings.