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 $f$ is analytic and injective in the punctured
open unit disk
$\mathbb D\setminus\$ and has the power series representation
:$f(z)= \frac 1z + \sum_^\infty a_n z^n,\qquad z\in \mathbb D\setminus\,$
then the coefficients $a_n$ satisfy
:$\sum_^\infty n, a_n, ^2\le 1.$

Proof

The idea of the proof is to look at the area uncovered by the image of $f$.
Define for $r\in(0,1)$
:\gamma_r(\theta):=f(r\,e^),\qquad \theta\in ,2\pi

Then $\gamma_r$ is a simple closed curve in the plane.
Let $D_r$ denote the unique bounded connected component of
\mathbb C\setminus\gamma_r( ,2\pi. The existence and
uniqueness of $D_r$ follows from Jordan's curve theorem.

If $D$ is a domain in the plane whose boundary
is a smooth simple closed curve $\gamma$,
then
:$\mathrm(D)=\int_\gamma x\,dy=-\int_\gamma y\,dx\,,$
provided that $\gamma$ is positively oriented
around $D$.
This follows easily, for example, from Green's theorem.
As we will soon see, $\gamma_r$ is positively oriented around
$D_r$ (and that is the reason for the minus sign in the
definition of $\gamma_r$). After applying the chain rule
and the formula for $\gamma_r$, the above expressions for
the area give
:$\mathrm(D_r)= \int_0^ \Re\bigl(f(r e^)\bigr)\,\Im\bigl(-i\,r\,e^\,f'(r e^)\bigr)\,d\theta = -\int_0^ \Im\bigl(f(r e^)\bigr)\,\Re\bigl(-i\,r\,e^\,f'(r e^)\bigr)d\theta.$
Therefore, the area of $D_r$ also equals to the average of the two expressions on the right
hand side. After simplification, this yields
:$\mathrm(D_r) = -\frac 12\, \Re\int_0^f(r\,e^)\,\overline\,d\theta,$
where $\overline z$ denotes complex conjugation. We set $a_=1$ and use the power series
expansion for $f$, to get
:$\mathrm(D_r) = -\frac 12\, \Re\int_0^ \sum_^\infty \sum_^\infty m\,r^\,a_n\,\overline\,e^\,d\theta\,.$
(Since $\int_0^ \sum_^\infty\sum_^\infty m\,r^\,, a_n, \,, a_m, \,d\theta<\infty\,,$ the rearrangement of the terms is justified.)
Now note that $\int_0^ e^\,d\theta$ is $2\pi$ if $n= m$
and is zero otherwise. Therefore, we get
:$\mathrm(D_r)= -\pi\sum_^\infty n\,r^\,, a_n, ^2.$
The area of $D_r$ is clearly positive. Therefore, the right hand side
is positive. Since $a_=1$, by letting $r\to1$, the
theorem now follows.

It only remains to justify the claim that $\gamma_r$ is positively oriented
around $D_r$. Let $r'$ satisfy r, and set
$z_0=f(r')$, say. For very small $s>0$, we may write the
expression for the winding number of $\gamma_s$ around $z_0$,
and verify that it is equal to $1$. Since, $\gamma_t$ does
not pass through $z_0$ when $t\ne r'$
(as $f$ is injective), the invariance
of the winding number under homotopy in the complement of $z_0$
implies that the winding number of
$\gamma_r$ around $z_0$ is also $1$.
This implies that $z_0\in D_r$ and that $\gamma_r$
is positively oriented around $D_r$, 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.

References

*{{Citation , last1=Rudin , first1=Walter , author1-link=Walter Rudin , title=Real and complex analysis , publisher=McGraw-Hill Book Co. , location=New York , edition=3rd , isbn=978-0-07-054234-1 , mr=924157 , year=1987 , oclc=13093736

Theorems in complex analysis
Articles containing proofs