Area theorem (conformal mapping) explained

In the mathematical theory of conformal mappings, the area theoremgives an inequality satisfied bythe power series coefficients of certain conformal mappings.The theorem is called by that name, not because of its implications, but rather because the proof usesthe notion of area.

Statement

Suppose that

f is analytic and injective in the puncturedopen unit diskD\setminus\{0\} and has the power series representationf(z) = (1 z)/(+) \sum ^{inf ty} _{n =0} a _{n} z ^{n,} z\in D\setminus\{0\}, then the coefficients a _{n} satisfy\sum ^{inf ty} _{n =0} n|a ^{2 \le} _{n|} 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 ^{- i\theta} _{r(\theta): = f(re} ), \theta\in[ 0 , 2 \pi]. Then \gamma _{r} is a simple closed curve in the plane.Let D _{r} denote the unique bounded connected component ofC\setminus\gamma _{r([0,2\pi])} . The existence anduniqueness of D _{r} follows from Jordan's curve theorem.

If

D is a domain in the plane whose boundaryis a smooth simple closed curve \gamma ,thenarea (D) = \int _{\gamma} xdy =- \int _{\gamma} ydx, provided that \gamma is positively orientedaround D .This follows easily, for example, from Green's theorem.As we will soon see, \gamma _{r} is positively oriented aroundD _{r} (and that is the reason for the minus sign in thedefinition of \gamma _{r} ). After applying the chain ruleand the formula for \gamma _{r} , the above expressions forthe area givearea (D _{r)=} \int ^{2 \pi} _{0} \Re l (f(r e ^{-i\theta} ) r )\Im l ( - ire ^{-i\theta} f'(r e ^{-i\theta} ) r )d\theta = - \int ^{2 \pi} _{0} \Im l (f(r e ^{-i\theta} ) r )\Re l ( - ire ^{-i\theta} f'(r e ^{-i\theta} ) r )d\theta. Therefore, the area of D _{r} also equals to the average of the two expressions on the righthand side. After simplification, this yieldsarea (D _{r)} =- (12)/(\Re\int) ^{2 \pi} ^{0} f(re ^{-i\theta} )\overline{re ^{-i\theta} f'(re ^{-i\theta} )}d\theta, where \overline z denotes complex conjugation. We set a _{-1} =1 and use the power seriesexpansion for f , to getarea (D _{r)} =- (12)/(\Re\int) ^{2 \pi} ^{0} \sum ^{inf ty mr} _{m =-1} ^{n+m} a _{n\overline{a} ^{i(m - n)\theta} ^{m}e} d\theta. (Since \int ^{2 \pi} _{0} \sum ^{inf ty} _{m =-1} mr ^{n+m} |a _{n||a} _{m|d\theta<infty,} the rearrangement of the terms is justified.)Now note that \int ^{2 \pi} _{0} e ^{i(m-n)\theta} d\theta is 2 \pi if n = m and is zero otherwise. Therefore, we getarea (D _{r)=} - \pi\sum ^{inf ty} _{n =-1} nr ^{2n} |a ^{2 .} _{n|} The area of D _{r} is clearly positive. Therefore, the right hand sideis positive. Since a _{-1} =1 , by letting r\to 1 , thetheorem now follows.

It only remains to justify the claim that

\gamma _{r} is positively orientedaround D _{r} . Let r' satisfy r<r'< 1 , and setz _{0=f(r')} , say. For very small s> 0 , we may write theexpression for the winding number of \gamma _{s} around z _{0} ,and verify that it is equal to 1 . Since, \gamma _{t} doesnot pass through z _{0} when t\ne r' (as f is injective), the invarianceof 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 conformalmappings were of considerable interest to mathematicians prior tothe solution of the Bieberbach conjecture. The area theoremis a central tool in this context. Moreover, the area theorem is oftenused in order to prove the Koebe 1/4 theorem, which is veryuseful in the study of the geometry of conformal mappings.