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Koebe quarter theorem
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<h2 id="grnwalls-area-theorem">Grönwall's area theorem</h2> <p class="note">Main article: <a href="/facts/Area_theorem_(conformal_mapping)/7cyDxcQu">Area theorem (conformal mapping)</a></p> <p>Suppose that </p> g ( z ) = z + b 1 z − 1 + b 2 z − 2 + ⋯ {\displaystyle g(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots } <p>is univalent in | z | > 1 {\displaystyle |z|>1} . Then </p> ∑ n = 1 ∞ n | b n | 2 ≤ 1. {\displaystyle \sum _{n=1}^{\infty }n|b_{n}|^{2}\leq 1.} <p>In fact, if r > 1 {\displaystyle r>1} , the complement of the image of the disk | z | > r {\displaystyle |z|>r} is a bounded domain X ( r ) {\displaystyle X(r)} . Its area is given by </p> ∫ X ( r ) d x d y = 1 2 i ∫ ∂ X ( r ) z ¯ d z = 1 2 i ∫ | z | = r g ¯ d g = π r 2 − π ∑ n = 1 ∞ n | b n | 2 r − 2 n . {\displaystyle \int _{X(r)}dx\,dy={1 \over 2i}\int _{\partial X(r)}{\overline {z}}\,dz={1 \over 2i}\int _{|z|=r}{\overline {g}}\,dg=\pi r^{2}-\pi \sum _{n=1}^{\infty }n|b_{n}|^{2}r^{-2n}.} <p>Since the area is positive, the result follows by letting r {\displaystyle r} decrease to 1 {\displaystyle 1} . The above proof shows equality holds if and only if the complement of the image of g {\displaystyle g} has zero area, i.e. <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a> zero. </p><p>This result was proved in 1914 by the Swedish mathematician <a href="/facts/Thomas_Hakon_Gr%25C3%25B6nwall/U9dqtxbJ">Thomas Hakon Grönwall</a>. </p> <h2 id="koebe-function">Koebe function</h2> <p>The Koebe function is defined by </p> f ( z ) = z ( 1 − z ) 2 = ∑ n = 1 ∞ n z n {\displaystyle f(z)={\frac {z}{(1-z)^{2}}}=\sum _{n=1}^{\infty }nz^{n}} <p>Application of the theorem to this function shows that the constant 1 / 4 {\displaystyle 1/4} in the theorem cannot be improved, as the image domain f ( D ) {\displaystyle f(\mathbf {D} )} does not contain the point z = − 1 / 4 {\displaystyle z=-1/4} and so cannot contain any disk centred at 0 {\displaystyle 0} with radius larger than 1 / 4 {\displaystyle 1/4} . </p><p>The rotated Koebe function is </p> f α ( z ) = z ( 1 − α z ) 2 = ∑ n = 1 ∞ n α n − 1 z n {\displaystyle f_{\alpha }(z)={\frac {z}{(1-\alpha z)^{2}}}=\sum _{n=1}^{\infty }n\alpha ^{n-1}z^{n}} <p>with α {\displaystyle \alpha } a <a href="/facts/Complex_number/2w7mqI7e">complex number</a> of <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> 1 {\displaystyle 1} . The Koebe function and its rotations are <i><a href="/facts/Schlicht_function/PXKcc8un">schlicht</a></i>: that is, <a href="/facts/Univalent_function/fQc4E6Ju">univalent</a> (analytic and <a href="/facts/Injective_function/5ES6tqf5">one-to-one</a>) and satisfying f ( 0 ) = 0 {\displaystyle f(0)=0} and f ′ ( 0 ) = 1 {\displaystyle f'(0)=1} . </p> <h2 id="bieberbachs-coefficient-inequality-for-univalent-functions">Bieberbach's coefficient inequality for univalent functions</h2> <p>Let </p> g ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ {\displaystyle g(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots } <p>be univalent in | z | < 1 {\displaystyle |z|<1} . Then </p> | a 2 | ≤ 2. {\displaystyle |a_{2}|\leq 2.} <p>This follows by applying Gronwall's area theorem to the odd univalent function </p> g ( z − 2 ) − 1 / 2 = z − 1 2 a 2 z − 1 + ⋯ . {\displaystyle g(z^{-2})^{-1/2}=z-{1 \over 2}a_{2}z^{-1}+\cdots .} <p>Equality holds if and only if g {\displaystyle g} is a rotated Koebe function. </p><p>This result was proved by <a href="/facts/Ludwig_Bieberbach/Ke12coBy">Ludwig Bieberbach</a> in 1916 and provided the basis for his <a href="/facts/Bieberbach_conjecture/PXKcc8un">celebrated conjecture</a> that | a n | ≤ n {\displaystyle |a_{n}|\leq n} , proved in 1985 by <a href="/facts/Louis_de_Branges/CJmL8SYU">Louis de Branges</a>. </p> <h2 id="proof-of-quarter-theorem">Proof of quarter theorem</h2> <p>Applying an affine map, it can be assumed that </p> f ( 0 ) = 0 , f ′ ( 0 ) = 1 , {\displaystyle f(0)=0,\,\,\,f^{\prime }(0)=1,} <p>so that </p> f ( z ) = z + a 2 z 2 + ⋯ . {\displaystyle f(z)=z+a_{2}z^{2}+\cdots .} <p>In particular, the coefficient inequality gives that | a 2 | ≤ 2 {\displaystyle |a_{2}|\leq 2} . If w {\displaystyle w} is not in f ( D ) {\displaystyle f(\mathbf {D} )} , then </p> h ( z ) = w f ( z ) w − f ( z ) = z + ( a 2 + w − 1 ) z 2 + ⋯ {\displaystyle h(z)={wf(z) \over w-f(z)}=z+(a_{2}+w^{-1})z^{2}+\cdots } <p>is univalent in | z | < 1 {\displaystyle |z|<1} . </p><p>Applying the coefficient inequality to h {\displaystyle h} gives </p> | w | − 1 = | w − 1 | = | − a 2 + a 2 + w − 1 | ≤ | a 2 | + | a 2 + w − 1 | ≤ 4 , {\displaystyle |w|^{-1}=|w^{-1}|=|-a_{2}+a_{2}+w^{-1}|\leq |a_{2}|+|a_{2}+w^{-1}|\leq 4,} <p>so that </p> | w | ≥ 1 4 . {\displaystyle |w|\geq {1 \over 4}.} <h2 id="koebe-distortion-theorem">Koebe distortion theorem</h2> <p>The Koebe distortion theorem gives a series of bounds for a univalent function and its derivative. It is a direct consequence of Bieberbach's inequality for the second coefficient and the Koebe quarter theorem.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p><p>Let f ( z ) {\displaystyle f(z)} be a univalent function on | z | < 1 {\displaystyle |z|<1} normalized so that f ( 0 ) = 0 {\displaystyle f(0)=0} and f ′ ( 0 ) = 1 {\displaystyle f'(0)=1} and let r = | z | {\displaystyle r=|z|} . Then </p> r ( 1 + r ) 2 ≤ | f ( z ) | ≤ r ( 1 − r ) 2 {\displaystyle {r \over (1+r)^{2}}\leq |f(z)|\leq {r \over (1-r)^{2}}} 1 − r ( 1 + r ) 3 ≤ | f ′ ( z ) | ≤ 1 + r ( 1 − r ) 3 {\displaystyle {1-r \over (1+r)^{3}}\leq |f^{\prime }(z)|\leq {1+r \over (1-r)^{3}}} 1 − r 1 + r ≤ | z f ′ ( z ) f ( z ) | ≤ 1 + r 1 − r {\displaystyle {1-r \over 1+r}\leq \left|z{f^{\prime }(z) \over f(z)}\right|\leq {1+r \over 1-r}} <p>with equality if and only if f {\displaystyle f} is a Koebe function </p> f ( z ) = z ( 1 − e i θ z ) 2 . {\displaystyle f(z)={z \over (1-e^{i\theta }z)^{2}}.} <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Ludwig_Bieberbach/Ke12coBy">Bieberbach, Ludwig</a> (1916), "Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln", <i>S.-B. Preuss. Akad. Wiss.</i>: 940–955</li> <li><a href="/facts/Lennart_Carleson/drIokpo0">Carleson, L.</a>; Gamelin, T. D. W. (1993), <a href="https://archive.org/details/complexdynamics0000carl/page/1"><i>Complex dynamics</i></a>, Universitext: Tracts in Mathematics, Springer-Verlag, pp. <a href="https://archive.org/details/complexdynamics0000carl/page/1">1–2</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-97942-5</li> <li><a href="/facts/John_B._Conway/7aMjnAxr">Conway, John B.</a> (1995), <i>Functions of One Complex Variable II</i>, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94460-9</li> <li><a href="/facts/Peter_Duren/had6hlV6">Duren, P. L.</a> (1983), <i>Univalent functions</i>, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90795-5</li> <li><a href="/facts/Thomas_Hakon_Gr%25C3%25B6nwall/U9dqtxbJ">Gronwall, T.H.</a> (1914), "Some remarks on conformal representation", <i><a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a></i>, 16 (1/4): 72–76, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1968044">10.2307/1968044</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1968044">1968044</a></li> <li><a href="/facts/Zeev_Nehari/Sc3fMc4k">Nehari, Zeev</a> (1952), <a href="https://archive.org/details/conformalmapping00neha/page/248"><i>Conformal mapping</i></a>, Dover, pp. <a href="https://archive.org/details/conformalmapping00neha/page/248">248–249</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-61137-X {{citation}}: ISBN / Date incompatibility (help)</li> <li><a href="/facts/Christian_Pommerenke/7xpHq182">Pommerenke, C.</a> (1975), <i>Univalent functions, with a chapter on quadratic differentials by Gerd Jensen</i>, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht</li> <li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1987). <i>Real and Complex Analysis</i>. Series in Higher Mathematics (3 ed.). McGraw-Hill. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-054234-1. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0924157">0924157</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li>Koebe 1/4 theorem at <a href="https://planetmath.org/koebe14theorem">PlanetMath</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Pommerenke 1975, pp. 21–22 - Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>