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Area theorem (conformal mapping)
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<h2 id="statement">Statement</h2> <p>Suppose that f {\displaystyle f} is <a href="/facts/Analytic_function/JhL3z8FP">analytic</a> and <a href="/facts/Injective_function/5ES6tqf5">injective</a> in the punctured <a href="/facts/Open_set/QfTCIqMu">open</a> <a href="/facts/Unit_disk/phrUu7yC">unit disk</a> D ∖ { 0 } {\displaystyle \mathbb {D} \setminus \{0\}} and has the power series representation </p> f ( z ) = 1 z + ∑ n = 0 ∞ a n z n , z ∈ D ∖ { 0 } , {\displaystyle f(z)={\frac {1}{z}}+\sum _{n=0}^{\infty }a_{n}z^{n},\qquad z\in \mathbb {D} \setminus \{0\},} <p>then the coefficients a n {\displaystyle a_{n}} satisfy </p> ∑ n = 0 ∞ n | a n | 2 ≤ 1. {\displaystyle \sum _{n=0}^{\infty }n|a_{n}|^{2}\leq 1.} <h2 id="proof">Proof</h2> <p>The idea of the proof is to look at the area uncovered by the image of f {\displaystyle f} . Define for r ∈ ( 0 , 1 ) {\displaystyle r\in (0,1)} </p> γ r ( θ ) := f ( r e − i θ ) , θ ∈ [ 0 , 2 π ] . {\displaystyle \gamma _{r}(\theta ):=f(r\,e^{-i\theta }),\qquad \theta \in [0,2\pi ].} <p>Then γ r {\displaystyle \gamma _{r}} is a simple closed curve in the plane. Let D r {\displaystyle D_{r}} denote the unique bounded connected component of C ∖ γ r ( [ 0 , 2 π ] ) {\displaystyle \mathbb {C} \setminus \gamma _{r}([0,2\pi ])} . The existence and uniqueness of D r {\displaystyle D_{r}} follows from <a href="/facts/Jordan_curve_theorem/zKpty4QT">Jordan's curve theorem</a>. </p><p>If D {\displaystyle D} is a domain in the plane whose boundary is a <a href="/facts/Smooth_function/mffL4ch2">smooth</a> simple closed curve γ {\displaystyle \gamma } , then </p> a r e a ( D ) = ∫ γ x d y = − ∫ γ y d x , {\displaystyle \mathrm {area} (D)=\int _{\gamma }x\,dy=-\int _{\gamma }y\,dx\,,} <p>provided that γ {\displaystyle \gamma } is positively <a href="/facts/Orientation_(topology)/1SEQM2Pt">oriented</a> around D {\displaystyle D} . This follows easily, for example, from <a href="/facts/Green%2527s_theorem/WBDk654A">Green's theorem</a>. As we will soon see, γ r {\displaystyle \gamma _{r}} is positively oriented around D r {\displaystyle D_{r}} (and that is the reason for the minus sign in the definition of γ r {\displaystyle \gamma _{r}} ). After applying the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a> and the formula for γ r {\displaystyle \gamma _{r}} , the above expressions for the area give </p> a r e a ( D r ) = ∫ 0 2 π ℜ ( f ( r e − i θ ) ) ℑ ( − i r e − i θ f ′ ( r e − i θ ) ) d θ = − ∫ 0 2 π ℑ ( f ( r e − i θ ) ) ℜ ( − i r e − i θ f ′ ( r e − i θ ) ) d θ . {\displaystyle \mathrm {area} (D_{r})=\int _{0}^{2\pi }\Re {\bigl (}f(re^{-i\theta }){\bigr )}\,\Im {\bigl (}-i\,r\,e^{-i\theta }\,f'(re^{-i\theta }){\bigr )}\,d\theta =-\int _{0}^{2\pi }\Im {\bigl (}f(re^{-i\theta }){\bigr )}\,\Re {\bigl (}-i\,r\,e^{-i\theta }\,f'(re^{-i\theta }){\bigr )}d\theta .} <p>Therefore, the area of D r {\displaystyle D_{r}} also equals to the average of the two expressions on the right hand side. After simplification, this yields </p> a r e a ( D r ) = − 1 2 ℜ ∫ 0 2 π f ( r e − i θ ) r e − i θ f ′ ( r e − i θ ) ¯ d θ , {\displaystyle \mathrm {area} (D_{r})=-{\frac {1}{2}}\,\Re \int _{0}^{2\pi }f(r\,e^{-i\theta })\,{\overline {r\,e^{-i\theta }\,f'(r\,e^{-i\theta })}}\,d\theta ,} <p>where z ¯ {\displaystyle {\overline {z}}} denotes <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a>. We set a − 1 = 1 {\displaystyle a_{-1}=1} and use the power series expansion for f {\displaystyle f} , to get </p> a r e a ( D r ) = − 1 2 ℜ ∫ 0 2 π ∑ n = − 1 ∞ ∑ m = − 1 ∞ m r n + m a n a m ¯ e i ( m − n ) θ d θ . {\displaystyle \mathrm {area} (D_{r})=-{\frac {1}{2}}\,\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 \,.} <p>(Since ∫ 0 2 π ∑ n = − 1 ∞ ∑ m = − 1 ∞ m r n + m | a n | | a m | d θ < ∞ , {\displaystyle \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 ∫ 0 2 π e i ( m − n ) θ d θ {\displaystyle \int _{0}^{2\pi }e^{i\,(m-n)\,\theta }\,d\theta } is 2 π {\displaystyle 2\pi } if n = m {\displaystyle n=m} and is zero otherwise. Therefore, we get </p> a r e a ( D r ) = − π ∑ n = − 1 ∞ n r 2 n | a n | 2 . {\displaystyle \mathrm {area} (D_{r})=-\pi \sum _{n=-1}^{\infty }n\,r^{2n}\,|a_{n}|^{2}.} <p>The area of D r {\displaystyle D_{r}} is clearly positive. Therefore, the right hand side is positive. Since a − 1 = 1 {\displaystyle a_{-1}=1} , by letting r → 1 {\displaystyle r\to 1} , the theorem now follows. </p><p>It only remains to justify the claim that γ r {\displaystyle \gamma _{r}} is positively oriented around D r {\displaystyle D_{r}} . Let r ′ {\displaystyle r'} satisfy r < r ′ < 1 {\displaystyle r<r'<1} , and set z 0 = f ( r ′ ) {\displaystyle z_{0}=f(r')} , say. For very small s > 0 {\displaystyle s>0} , we may write the expression for the <a href="/facts/Winding_number/kaquyUL2">winding number</a> of γ s {\displaystyle \gamma _{s}} around z 0 {\displaystyle z_{0}} , and verify that it is equal to 1 {\displaystyle 1} . Since, γ t {\displaystyle \gamma _{t}} does not pass through z 0 {\displaystyle z_{0}} when t ≠ r ′ {\displaystyle t\neq r'} (as f {\displaystyle f} is injective), the invariance of the winding number under homotopy in the complement of z 0 {\displaystyle z_{0}} implies that the winding number of γ r {\displaystyle \gamma _{r}} around z 0 {\displaystyle z_{0}} is also 1 {\displaystyle 1} . This implies that z 0 ∈ D r {\displaystyle z_{0}\in D_{r}} and that γ r {\displaystyle \gamma _{r}} is positively oriented around D r {\displaystyle D_{r}} , as required. </p> <h2 id="uses">Uses</h2> <p>The inequalities satisfied by power series coefficients of conformal mappings were of considerable interest to mathematicians prior to the solution of the <a href="/facts/Bieberbach_conjecture/PXKcc8un">Bieberbach conjecture</a>. The area theorem is a central tool in this context. Moreover, the area theorem is often used in order to prove the <a href="/facts/Koebe_1%2f4_theorem/IAyU9SWC">Koebe 1/4 theorem</a>, which is very useful in the study of the geometry of conformal mappings. </p> <ul><li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1987), <i>Real and complex analysis</i> (3rd ed.), New York: McGraw-Hill Book Co., <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054234-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0924157">0924157</a>, <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/13093736">13093736</a></li></ul>