Area theorem (conformal mapping)

In the mathematical theory of conformal mappings, the area theorem gives 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.

tatement

Suppose that $f$ is analytic and injective in the punctured
open unit disk$mathbb\; Dsetminus\{0\}$ and has the power series representation:$f(z)=\; frac\; 1z\; +\; sum\_\{n=0\}^infty\; a\_n\; z^n,qquad\; zin\; mathbb\; Dsetminus\{0\},$then the coefficients $a\_n$ satisfy:$sum\_\{n=0\}^infty\; n|a\_n|^2le\; 1.$

Proof

The idea of the proof is to look at the area uncovered by the image of $f$.Define for $rin(0,1)$:$gamma\_r(\; heta):=f(r,e^\{-i\; heta\}),qquad\; hetain\; [0,2pi]\; .$Then $gamma\_r$ is a simple closed curve in the plane.Let $D\_r$ denote the unique bounded connected component of$mathbb\; Csetminusgamma\; [0,2pi]$. 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 $gamma$,then:$mathrm\{area\}(D)=int\_gamma\; x,dy=-int\_gamma\; y,dx,,$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 thedefinition of $gamma\_r$). After applying the chain ruleand the formula for $gamma\_r$, the above expressions forthe area give:Therefore, the area of $D\_r$ also equals to the average of the two expressions on the righthand side. After simplification, this yields:$mathrm\{area\}(D\_r)\; =\; -frac\; 12,\; Reint\_0^\{2pi\}f(r,e^\{-i\; heta\}),overline\{r,e^\{-i\; heta\},f\text{'}(r,e^\{-i\; heta\})\},d\; heta,$where $overline\; z$ denotes complex conjugation. We set $a\_\{-1\}=1$ and use the power seriesexpansion for $f$, to get:$mathrm\{area\}(D\_r)\; =\; -frac\; 12,\; Reint\_0^\{2pi\}\; sum\_\{n=-1\}^inftysum\_\{m=-1\}^inftym,r^\{n+m\},a\_n,overline\{a\_m\},e^\{i,(m-n),\; heta\},d\; heta,.$(Since

It only remains to justify the claim that $gamma\_r$ is positively orientedaround $D\_r$. Let $r\text{'}$ satisfy

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.

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 | id=MathSciNet | id = 924157 | year=1987