- Area theorem (conformal mapping)
In the mathematical theory of
conformal mapping s, the area theorem gives an inequality satisfied bythepower series coefficient s of certain conformal mappings. The theorem is called by that name, not because of its implications, but rather because the proof usesthe notion ofarea .tatement
Suppose that f is analytic and injective in the punctured
openunit 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 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 rule and 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 denotescomplex 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 hetathe 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 theKoebe 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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