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.

Suppose that ${\displaystyle f}$ is analytic and injective in the punctured open unit disk ${\displaystyle \mathbb {D} \setminus \{0\}}$ and has the power series representation

${\displaystyle f(z)={\frac {1}{z}}+\sum _{n=0}^{\infty }a_{n}z^{n},\qquad z\in \mathbb {D} \setminus \{0\},}$

then the coefficients ${\displaystyle a_{n}}$ satisfy

${\displaystyle \sum _{n=0}^{\infty }n|a_{n}|^{2}\leq 1.}$

The idea of the proof is to look at the area uncovered by the image of ${\displaystyle f}$. Define for ${\displaystyle r\in (0,1)}$

${\displaystyle \gamma _{r}(\theta ):=f(r\,e^{-i\theta }),\qquad \theta \in [0,2\pi ].}$

Then ${\displaystyle \gamma _{r}}$ is a simple closed curve in the plane. Let ${\displaystyle D_{r}}$ denote the unique bounded connected component of ${\displaystyle \mathbb {C} \setminus \gamma _{r}([0,2\pi ])}$. The existence and uniqueness of ${\displaystyle D_{r}}$ follows from Jordan's curve theorem.

If ${\displaystyle D}$ is a domain in the plane whose boundary is a smooth simple closed curve ${\displaystyle \gamma }$, then

${\displaystyle \mathrm {area} (D)=\int _{\gamma }x\,dy=-\int _{\gamma }y\,dx\,,}$

provided that ${\displaystyle \gamma }$ is positively oriented around ${\displaystyle D}$. This follows easily, for example, from Green's theorem. As we will soon see, ${\displaystyle \gamma _{r}}$ is positively oriented around ${\displaystyle D_{r}}$ (and that is the reason for the minus sign in the definition of ${\displaystyle \gamma _{r}}$). After applying the chain rule and the formula for ${\displaystyle \gamma _{r}}$, the above expressions for the area give

${\displaystyle \mathrm {area} (D_{r})=\int _{0}^{2\pi }\Re {\bigl (}f(re^{-i\theta }){\bigr )}\,\Im {\bigl (}-i\,r\,e^{-i\theta }\,f'(re^{-i\theta }){\bigr )}\,d\theta =-\int _{0}^{2\pi }\Im {\bigl (}f(re^{-i\theta }){\bigr )}\,\Re {\bigl (}-i\,r\,e^{-i\theta }\,f'(re^{-i\theta }){\bigr )}d\theta .}$

Therefore, the area of ${\displaystyle D_{r}}$ also equals to the average of the two expressions on the right hand side. After simplification, this yields

${\displaystyle \mathrm {area} (D_{r})=-{\frac {1}{2}}\,\Re \int _{0}^{2\pi }f(r\,e^{-i\theta })\,{\overline {r\,e^{-i\theta }\,f'(r\,e^{-i\theta })}}\,d\theta ,}$

where ${\displaystyle {\overline {z}}}$ denotes complex conjugation. We set ${\displaystyle a_{-1}=1}$ and use the power series expansion for ${\displaystyle f}$, to get

${\displaystyle \mathrm {area} (D_{r})=-{\frac {1}{2}}\,\Re \int _{0}^{2\pi }\sum _{n=-1}^{\infty }\sum _{m=-1}^{\infty }m\,r^{n+m}\,a_{n}\,{\overline {a_{m}}}\,e^{i\,(m-n)\,\theta }\,d\theta \,.}$

(Since ${\displaystyle \int _{0}^{2\pi }\sum _{n=-1}^{\infty }\sum _{m=-1}^{\infty }m\,r^{n+m}\,|a_{n}|\,|a_{m}|\,d\theta <\infty \,,}$ the rearrangement of the terms is justified.) Now note that ${\displaystyle \int _{0}^{2\pi }e^{i\,(m-n)\,\theta }\,d\theta }$ is ${\displaystyle 2\pi }$ if ${\displaystyle n=m}$ and is zero otherwise. Therefore, we get

${\displaystyle \mathrm {area} (D_{r})=-\pi \sum _{n=-1}^{\infty }n\,r^{2n}\,|a_{n}|^{2}.}$

The area of ${\displaystyle D_{r}}$ is clearly positive. Therefore, the right hand side is positive. Since ${\displaystyle a_{-1}=1}$, by letting ${\displaystyle r\to 1}$, the theorem now follows.

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

• Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill Book Co., ISBN 978-0-07-054234-1, MR 0924157, OCLC 13093736