Area theorem (conformal mapping)

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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.

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[edit] Statement

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

f(z)= \frac 1z + \sum_{n=0}^\infty a_n z^n,\qquad z\in \mathbb D\setminus\{0\},

then the coefficients an satisfy

\sum_{n=0}^\infty n|a_n|^2\le 1.

[edit] 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^{-i\theta}),\qquad \theta\in[0,2\pi].

Then γr is a simple closed curve in the plane. Let Dr denote the unique bounded connected component of \mathbb C\setminus\gamma[0,2\pi]. The existence and uniqueness of Dr follows from Jordan's curve theorem.

If D is a domain in the plane whose boundary is a smooth simple closed curve γ, then

\mathrm{area}(D)=\int_\gamma x\,dy=-\int_\gamma y\,dx\,,

provided that γ is positively oriented around D. This follows easily, for example, from Green's theorem. As we will soon see, γr is positively oriented around Dr (and that is the reason for the minus sign in the definition of γr). After applying the chain rule and the formula for γr, the above expressions for the area give

\mathrm{area}(D_r)= \int_0^{2\pi} \Re\bigl(f(r e^{-i\theta})\bigr)\,\Im\bigl(-i\,r\,e^{-i\theta}\,f'(r e^{-i\theta})\bigr)\,d\theta = -\int_0^{2\pi} \Im\bigl(f(r e^{-i\theta})\bigr)\,\Re\bigl(-i\,r\,e^{-i\theta}\,f'(r e^{-i\theta})\bigr).

Therefore, the area of Dr also equals to the average of the two expressions on the right hand side. After simplification, this yields

\mathrm{area}(D_r) = -\frac 12\, \Re\int_0^{2\pi}f(r\,e^{-i\theta})\,\overline{r\,e^{-i\theta}\,f'(r\,e^{-i\theta})}\,d\theta,

where \overline z denotes complex conjugation. We set a − 1 = 1 and use the power series expansion for f, to get

\mathrm{area}(D_r) = -\frac 12\, \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 \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 \int_0^{2\pi} e^{i\,(m-n)\,\theta}\,d\theta is 2π if n = m and is zero otherwise. Therefore, we get

\mathrm{area}(D_r)= -\pi\sum_{n=-1}^\infty n\,r^{2n}\,|a_n|^2.

The area of Dr is clearly positive. Therefore, the right hand side is positive. Since a − 1 = 1, by letting r\to1, the theorem now follows.

It only remains to justify the claim that γr is positively oriented around Dr. Let r' satisfy r < r' < 1, and set z0 = f(r'), say. For very small s > 0, we may write the expression for the winding number of γs around z0, and verify that it is equal to 1. Since, γt does not pass through z0 when t\ne r' (as f is injective), the invariance of the winding number under homotopy in the complement of z0 implies that the winding number of γr around z0 is also 1. This implies that z_0\in D_r and that γr is positively oriented around Dr, as required.

[edit] 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.

[edit] References