Area theorem (conformal mapping)

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 diskmathbb 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 ofmathbb 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 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 give:mathrm{area}(D_r)= int_0^{2pi} Reigl(f(r e^{-i heta})igr),Imigl(-i,r,e^{-i heta},f'(r e^{-i heta})igr),d heta = -int_0^{2pi} Imigl(f(r e^{-i heta})igr),Reigl(-i,r,e^{-i heta},f'(r e^{-i heta})igr).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'(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 int_0^{2pi} sum_{n=-1}^inftysum_{m=-1}^infty m,r^{n+m},|a_n|,|a_m|,d heta the rearrangement of the terms is justified.)Now note that int_0^{2pi} e^{i,(m-n), heta},d heta is 2pi if n= mand is zero otherwise. Therefore, we get:mathrm{area}(D_r)= -pisum_{n=-1}^infty n,r^{2n},|a_n|^2.The area of D_r is clearly positive. Therefore, the right hand sideis positive. Since a_{-1}=1, by letting r o1, thetheorem now follows.

It only remains to justify the claim that gamma_r is positively orientedaround D_r. Let r' satisfy r, 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 e r'(as f is injective), the invarianceof the winding number under homotopy in the complement of z_0 implies that the winding number ofgamma_r around z_0 is also 1.This implies that z_0in D_r and that gamma_ris 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.

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


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