<div dir="ltr"><div class="gmail_default" style="font-family:comic sans ms,sans-serif;font-size:large">---------- Forwarded message ----------<br></div><div class="gmail_quote">From: <b class="gmail_sendername">What's new</b> <span dir="ltr"><<a href="mailto:comment-reply@wordpress.com">comment-reply@wordpress.com</a>></span><br>Date: Thu, May 3, 2018 at 1:18 AM<br>Subject: [New post] 246C notes 3: Univalent functions, the Loewner equation, and the Bieberbach conjecture<br>To: <a href="mailto:yilmaz.akyildiz@gmail.com">yilmaz.akyildiz@gmail.com</a><br><br><a href="https://terrytao.wordpress.com/author/teorth/">https://terrytao.wordpress.com/author/teorth/</a></div><div class="gmail_quote"><br></div><div class="gmail_quote"><br><br><u></u>
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Terence Tao posted: "
We now approach conformal maps from yet another perspective. Given an open subset $latex {U}&fg=000000$ of the complex numbers $latex {{\bf C}}&fg=000000$, define a univalent function on $latex {U}&fg=000000$ to be a holomorphic function $l" </span>
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<h2 class="m_7396787748141865386post-title" style="margin:.4em 0 .3em;font-size:1.8em;font-size:1.6em;color:#555;margin:0;font-size:20px"><a href="https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/" style="text-decoration:underline;color:#2585b2;text-decoration:none!important" target="_blank">246C notes 3: Univalent functions, the Loewner equation, and the Bieberbach conjecture</a></h2>
<span style="color:#888">by <a href="https://terrytao.wordpress.com/author/teorth/" style="text-decoration:underline;color:#2585b2;color:#888!important" target="_blank">Terence Tao</a> </span>
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We now approach conformal maps from yet another perspective. Given an open subset <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0" alt="{U}" title="{U}" class="m_7396787748141865386latex"> of the complex numbers <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{{\bf C}}" title="{{\bf C}}" class="m_7396787748141865386latex">, define a <a style="text-decoration:underline;color:#2585b2" href="https://en.wikipedia.org/wiki/Univalent_function" target="_blank">univalent function</a> on <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0" alt="{U}" title="{U}" class="m_7396787748141865386latex"> to be a holomorphic function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%3A+U+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{f: U \rightarrow {\bf C}}" title="{f: U \rightarrow {\bf C}}" class="m_7396787748141865386latex"> that is also injective. We will primarily be studying this concept in the case when <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0" alt="{U}" title="{U}" class="m_7396787748141865386latex"> is the unit disk <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29+%3A%3D+%5C%7B+z+%5Cin+%7B%5Cbf+C%7D%3A+%7Cz%7C+%3C+1+%5C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1) := \{ z \in {\bf C}: |z| < 1 \}}" title="{D(0,1) := \{ z \in {\bf C}: |z| < 1 \}}" class="m_7396787748141865386latex">.
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Clearly, a univalent function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{f: D(0,1) \rightarrow {\bf C}}" title="{f: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex"> on the unit disk is a conformal map from <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex"> to the image <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%28D%280%2C1%29%29%7D&bg=ffffff&fg=000000&s=0" alt="{f(D(0,1))}" title="{f(D(0,1))}" class="m_7396787748141865386latex">; in particular, <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%28D%280%2C1%29%29%7D&bg=ffffff&fg=000000&s=0" alt="{f(D(0,1))}" title="{f(D(0,1))}" class="m_7396787748141865386latex"> is simply connected, and not all of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{{\bf C}}" title="{{\bf C}}" class="m_7396787748141865386latex"> (since otherwise the inverse map <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%5E%7B-1%7D%3A+%7B%5Cbf+C%7D+%5Crightarrow+D%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{f^{-1}: {\bf C} \rightarrow D(0,1)}" title="{f^{-1}: {\bf C} \rightarrow D(0,1)}" class="m_7396787748141865386latex"> would violate Liouville's theorem). In the converse direction, the Riemann mapping theorem tells us that every simply connected proper subset <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BV+%5Csubsetneq+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{V \subsetneq {\bf C}}" title="{V \subsetneq {\bf C}}" class="m_7396787748141865386latex"> of the complex numbers is the image of a univalent function on <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex">. Furthermore, if <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BV%7D&bg=ffffff&fg=000000&s=0" alt="{V}" title="{V}" class="m_7396787748141865386latex"> contains the origin, then the univalent function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{f: D(0,1) \rightarrow {\bf C}}" title="{f: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex"> with this image becomes unique once we normalise <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%280%29+%3D+0%7D&bg=ffffff&fg=000000&s=0" alt="{f(0) = 0}" title="{f(0) = 0}" class="m_7396787748141865386latex"> and <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%27%280%29+%3E+0%7D&bg=ffffff&fg=000000&s=0" alt="{f'(0) > 0}" title="{f'(0) > 0}" class="m_7396787748141865386latex">. Thus the Riemann mapping theorem provides a one-to-one correspondence between simply connected proper subsets of the complex plane containing the origin, and univalent functions <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{f: D(0,1) \rightarrow {\bf C}}" title="{f: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex"> with <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%280%29%3D0%7D&bg=ffffff&fg=000000&s=0" alt="{f(0)=0}" title="{f(0)=0}" class="m_7396787748141865386latex"> and <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%27%280%29%3E0%7D&bg=ffffff&fg=000000&s=0" alt="{f'(0)>0}" title="{f'(0)>0}" class="m_7396787748141865386latex">. We will focus particular attention on the univalent functions <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{f: D(0,1) \rightarrow {\bf C}}" title="{f: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex"> with the normalisation <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%280%29%3D0%7D&bg=ffffff&fg=000000&s=0" alt="{f(0)=0}" title="{f(0)=0}" class="m_7396787748141865386latex"> and <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%27%280%29%3D1%7D&bg=ffffff&fg=000000&s=0" alt="{f'(0)=1}" title="{f'(0)=1}" class="m_7396787748141865386latex">; such functions will be called <a style="text-decoration:underline;color:#2585b2" href="https://en.wikipedia.org/wiki/De_Branges%27s_theorem#Schlicht_functions" target="_blank">schlicht functions</a>.
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One basic example of a univalent function on <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex"> is the <a style="text-decoration:underline;color:#2585b2" href="https://en.wikipedia.org/wiki/Cayley_transform" target="_blank">Cayley transform</a> <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bz+%5Cmapsto+%5Cfrac%7B1%2Bz%7D%7B1-z%7D%7D&bg=ffffff&fg=000000&s=0" alt="{z \mapsto \frac{1+z}{1-z}}" title="{z \mapsto \frac{1+z}{1-z}}" class="m_7396787748141865386latex">, which is a Möbius transformation from <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex"> to the right half-plane <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%5C%7B+%5Cmathrm%7BRe%7D%28z%29+%3E+0+%5C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{\{ \mathrm{Re}(z) > 0 \}}" title="{\{ \mathrm{Re}(z) > 0 \}}" class="m_7396787748141865386latex">. (The slight variant <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bz+%5Cmapsto+%5Cfrac%7B1-z%7D%7B1%2Bz%7D%7D&bg=ffffff&fg=000000&s=0" alt="{z \mapsto \frac{1-z}{1+z}}" title="{z \mapsto \frac{1-z}{1+z}}" class="m_7396787748141865386latex"> is also referred to as the Cayley transform, as is the closely related map <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bz+%5Cmapsto+%5Cfrac%7Bz-i%7D%7Bz%2Bi%7D%7D&bg=ffffff&fg=000000&s=0" alt="{z \mapsto \frac{z-i}{z+i}}" title="{z \mapsto \frac{z-i}{z+i}}" class="m_7396787748141865386latex">, which maps <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex"> to the upper half-plane.) One can square this map to obtain a further univalent function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bz+%5Cmapsto+%5Cleft%28+%5Cfrac%7B1%2Bz%7D%7B1-z%7D+%5Cright%29%5E2%7D&bg=ffffff&fg=000000&s=0" alt="{z \mapsto \left( \frac{1+z}{1-z} \right)^2}" title="{z \mapsto \left( \frac{1+z}{1-z} \right)^2}" class="m_7396787748141865386latex">, which now maps <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex"> to the complex numbers with the negative real axis <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%28-%5Cinfty%2C0%5D%7D&bg=ffffff&fg=000000&s=0" alt="{(-\infty,0]}" title="{(-\infty,0]}" class="m_7396787748141865386latex"> removed. One can normalise this function to be schlicht to obtain the <a style="text-decoration:underline;color:#2585b2" href="https://www.encyclopediaofmath.org/index.php/Koebe_function" target="_blank">Koebe function</a> <a style="text-decoration:underline;color:#2585b2" name="m_7396787748141865386_koebe"></a></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++f%28z%29+%3A%3D+%5Cfrac%7B1%7D%7B4%7D%5Cleft%28+%5Cleft%28+%5Cfrac%7B1%2Bz%7D%7B1-z%7D+%5Cright%29%5E2+-+1%5Cright%29+%3D+%5Cfrac%7Bz%7D%7B%281-z%29%5E2%7D%2C+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0" alt="\displaystyle f(z) := \frac{1}{4}\left( \left( \frac{1+z}{1-z} \right)^2 - 1\right) = \frac{z}{(1-z)^2}, \ \ \ \ \ (1)" title="\displaystyle f(z) := \frac{1}{4}\left( \left( \frac{1+z}{1-z} \right)^2 - 1\right) = \frac{z}{(1-z)^2}, \ \ \ \ \ (1)" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> which now maps <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex"> to the complex numbers with the half-line <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%5B1%2F4%2C%5Cinfty%29%7D&bg=ffffff&fg=000000&s=0" alt="{[1/4,\infty)}" title="{[1/4,\infty)}" class="m_7396787748141865386latex"> removed. A little more generally, for any <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%5Ctheta+%5Cin+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0" alt="{\theta \in {\bf R}}" title="{\theta \in {\bf R}}" class="m_7396787748141865386latex"> we have the <em>rotated Koebe function</em> <a style="text-decoration:underline;color:#2585b2" name="m_7396787748141865386_rotated"></a></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++f%28z%29+%3A%3D+%5Cfrac%7Bz%7D%7B%281+-+e%5E%7Bi%5Ctheta%7D+z%29%5E2%7D+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0" alt="\displaystyle f(z) := \frac{z}{(1 - e^{i\theta} z)^2} \ \ \ \ \ (2)" title="\displaystyle f(z) := \frac{z}{(1 - e^{i\theta} z)^2} \ \ \ \ \ (2)" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> that is a schlicht function that maps <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex"> to the complex numbers with the half-line <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%5C%7B+re%5E%7B-i%5Ctheta%7D%3A+r+%5Cgeq+1%2F4%5C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{\{ re^{-i\theta}: r \geq 1/4\}}" title="{\{ re^{-i\theta}: r \geq 1/4\}}" class="m_7396787748141865386latex"> removed.
</p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em">
Every schlicht function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{f: D(0,1) \rightarrow {\bf C}}" title="{f: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex"> has a convergent Taylor expansion </p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++f%28z%29+%3D+a_1+z+%2B+a_2+z%5E2+%2B+a_3+z%5E3+%2B+%5Cdots&bg=ffffff&fg=000000&s=0" alt="\displaystyle f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots" title="\displaystyle f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> for some complex coefficients <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Ba_1%2Ca_2%2C%5Cdots%7D&bg=ffffff&fg=000000&s=0" alt="{a_1,a_2,\dots}" title="{a_1,a_2,\dots}" class="m_7396787748141865386latex"> with <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Ba_1%3D1%7D&bg=ffffff&fg=000000&s=0" alt="{a_1=1}" title="{a_1=1}" class="m_7396787748141865386latex">. For instance, the Koebe function has the expansion </p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++f%28z%29+%3D+z+%2B+2+z%5E2+%2B+3+z%5E3+%2B+%5Cdots+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+z%5En&bg=ffffff&fg=000000&s=0" alt="\displaystyle f(z) = z + 2 z^2 + 3 z^3 + \dots = \sum_{n=1}^\infty n z^n" title="\displaystyle f(z) = z + 2 z^2 + 3 z^3 + \dots = \sum_{n=1}^\infty n z^n" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> and similarly the rotated Koebe function has the expansion </p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++f%28z%29+%3D+z+%2B+2+e%5E%7Bi%5Ctheta%7D+z%5E2+%2B+3+e%5E%7B2i%5Ctheta%7D+z%5E3+%2B+%5Cdots+%3D+%5Csum_%7Bn%3D1%7D%5E%5Cinfty+n+e%5E%7B%28n-1%29%5Ctheta%7D+z%5En.&bg=ffffff&fg=000000&s=0" alt="\displaystyle f(z) = z + 2 e^{i\theta} z^2 + 3 e^{2i\theta} z^3 + \dots = \sum_{n=1}^\infty n e^{(n-1)\theta} z^n." title="\displaystyle f(z) = z + 2 e^{i\theta} z^2 + 3 e^{2i\theta} z^3 + \dots = \sum_{n=1}^\infty n e^{(n-1)\theta} z^n." class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> Intuitively, the Koebe function and its rotations should be the ``largest" schlicht functions available. This is formalised by the famous <a style="text-decoration:underline;color:#2585b2" href="https://en.wikipedia.org/wiki/De_Branges%27s_theorem" target="_blank">Bieberbach conjecture</a>, which asserts that for any schlicht function, the coefficients <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Ba_n%7D&bg=ffffff&fg=000000&s=0" alt="{a_n}" title="{a_n}" class="m_7396787748141865386latex"> should obey the bound <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%7Ca_n%7C+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0" alt="{|a_n| \leq n}" title="{|a_n| \leq n}" class="m_7396787748141865386latex"> for all <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0" alt="{n}" title="{n}" class="m_7396787748141865386latex">. After a large number of partial results, this conjecture was eventually <a style="text-decoration:underline;color:#2585b2" href="https://mathscinet.ams.org/mathscinet-getitem?mr=772434" target="_blank">solved by de Branges</a>; see for instance <a style="text-decoration:underline;color:#2585b2" href="https://mathscinet.ams.org/mathscinet-getitem?mr=856290" target="_blank">this survey of Korevaar</a> or <a style="text-decoration:underline;color:#2585b2" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2281158" target="_blank">this survey of Koepf</a> for a history.</p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em">
It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{g: D(0,1) \rightarrow {\bf C}}" title="{g: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex"> that are <em>odd</em>, thus <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%28-z%29%3D-g%28z%29%7D&bg=ffffff&fg=000000&s=0" alt="{g(-z)=-g(z)}" title="{g(-z)=-g(z)}" class="m_7396787748141865386latex"> for all <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bz%7D&bg=ffffff&fg=000000&s=0" alt="{z}" title="{z}" class="m_7396787748141865386latex">, and the Taylor expansion now reads </p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++g%28z%29+%3D+b_1+z+%2B+b_3+z%5E3+%2B+b_5+z%5E5+%2B+%5Cdots&bg=ffffff&fg=000000&s=0" alt="\displaystyle g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots" title="\displaystyle g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> for some complex coefficients <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bb_1%2Cb_3%2C%5Cdots%7D&bg=ffffff&fg=000000&s=0" alt="{b_1,b_3,\dots}" title="{b_1,b_3,\dots}" class="m_7396787748141865386latex"> with <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bb_1%3D1%7D&bg=ffffff&fg=000000&s=0" alt="{b_1=1}" title="{b_1=1}" class="m_7396787748141865386latex">. One can transform a general schlicht function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{f: D(0,1) \rightarrow {\bf C}}" title="{f: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex"> to an odd schlicht function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{g: D(0,1) \rightarrow {\bf C}}" title="{g: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex"> by observing that the function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%28z%5E2%29%2Fz%5E2%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{f(z^2)/z^2: D(0,1) \rightarrow {\bf C}}" title="{f(z^2)/z^2: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex">, after removing the singularity at zero, is a non-zero function that equals <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0" alt="{1}" title="{1}" class="m_7396787748141865386latex"> at the origin, and thus (as <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex"> is simply connected) has a unique holomorphic square root <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%28f%28z%5E2%29%2Fz%5E2%29%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0" alt="{(f(z^2)/z^2)^{1/2}}" title="{(f(z^2)/z^2)^{1/2}}" class="m_7396787748141865386latex"> that also equals <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0" alt="{1}" title="{1}" class="m_7396787748141865386latex"> at the origin. If one then sets <a style="text-decoration:underline;color:#2585b2" name="m_7396787748141865386_gsqr"></a></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++g%28z%29+%3A%3D+z+%28f%28z%5E2%29%2Fz%5E2%29%5E%7B1%2F2%7D+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0" alt="\displaystyle g(z) := z (f(z^2)/z^2)^{1/2} \ \ \ \ \ (3)" title="\displaystyle g(z) := z (f(z^2)/z^2)^{1/2} \ \ \ \ \ (3)" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> it is not difficult to verify that <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%7D&bg=ffffff&fg=000000&s=0" alt="{g}" title="{g}" class="m_7396787748141865386latex"> is an odd schlicht function which additionally obeys the equation <a style="text-decoration:underline;color:#2585b2" name="m_7396787748141865386_g-eq"></a></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++f%28z%5E2%29+%3D+g%28z%29%5E2.+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0" alt="\displaystyle f(z^2) = g(z)^2. \ \ \ \ \ (4)" title="\displaystyle f(z^2) = g(z)^2. \ \ \ \ \ (4)" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> Conversely, given an odd schlicht function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%7D&bg=ffffff&fg=000000&s=0" alt="{g}" title="{g}" class="m_7396787748141865386latex">, the formula <a style="text-decoration:underline;color:#2585b2" href="https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#g-eq" target="_blank">(4)</a> uniquely determines a schlicht function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0" alt="{f}" title="{f}" class="m_7396787748141865386latex">.</p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em">
For instance, if <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0" alt="{f}" title="{f}" class="m_7396787748141865386latex"> is the Koebe function <a style="text-decoration:underline;color:#2585b2" href="https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#koebe" target="_blank">(1)</a>, <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%7D&bg=ffffff&fg=000000&s=0" alt="{g}" title="{g}" class="m_7396787748141865386latex"> becomes <a style="text-decoration:underline;color:#2585b2" name="m_7396787748141865386_g-koebe"></a></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++g%28z%29+%3D+%5Cfrac%7Bz%7D%7B1-z%5E2%7D+%3D+z+%2B+z%5E3+%2B+z%5E5+%2B+%5Cdots%2C+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0" alt="\displaystyle g(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \dots, \ \ \ \ \ (5)" title="\displaystyle g(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \dots, \ \ \ \ \ (5)" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> which maps <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7BD%280%2C1%29%7D&bg=ffffff&fg=000000&s=0" alt="{D(0,1)}" title="{D(0,1)}" class="m_7396787748141865386latex"> to the complex numbers with two slits <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%5C%7B+%5Cpm+iy%3A+y+%3E+1%2F2+%5C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{\{ \pm iy: y > 1/2 \}}" title="{\{ \pm iy: y > 1/2 \}}" class="m_7396787748141865386latex"> removed, and if <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0" alt="{f}" title="{f}" class="m_7396787748141865386latex"> is the rotated Koebe function <a style="text-decoration:underline;color:#2585b2" href="https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#rotated" target="_blank">(2)</a>, <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%7D&bg=ffffff&fg=000000&s=0" alt="{g}" title="{g}" class="m_7396787748141865386latex"> becomes <a style="text-decoration:underline;color:#2585b2" name="m_7396787748141865386_g-koebe-rotated"></a></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++g%28z%29+%3D+%5Cfrac%7Bz%7D%7B1-+e%5E%7Bi%5Ctheta%7D+z%5E2%7D+%3D+z+%2B+e%5E%7Bi%5Ctheta%7D+z%5E3+%2B+e%5E%7B2i%5Ctheta%7D+z%5E5+%2B+%5Cdots.+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0" alt="\displaystyle g(z) = \frac{z}{1- e^{i\theta} z^2} = z + e^{i\theta} z^3 + e^{2i\theta} z^5 + \dots. \ \ \ \ \ (6)" title="\displaystyle g(z) = \frac{z}{1- e^{i\theta} z^2} = z + e^{i\theta} z^3 + e^{2i\theta} z^5 + \dots. \ \ \ \ \ (6)" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em">
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<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em">
De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as <a style="text-decoration:underline;color:#2585b2" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1503286" target="_blank">Robertson's conjecture</a>. More precisely, we have
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<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"><b>Theorem 1 (de Branges' theorem)</b> Let <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0" alt="{n \geq 1}" title="{n \geq 1}" class="m_7396787748141865386latex"> be a natural number. </p>
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<li style="line-height:1.6;margin-left:1em;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif">(i) (Robertson conjecture) If <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%28z%29+%3D+b_1+z+%2B+b_3+z%5E3+%2B+b_5+z%5E5+%2B+%5Cdots%7D&bg=ffffff&fg=000000&s=0" alt="{g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots}" title="{g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots}" class="m_7396787748141865386latex"> is an odd schlicht function, then
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk%3D1%7D%5En+%7Cb_%7B2k-1%7D%7C%5E2+%5Cleq+n.&bg=ffffff&fg=000000&s=0" alt="\displaystyle \sum_{k=1}^n |b_{2k-1}|^2 \leq n." title="\displaystyle \sum_{k=1}^n |b_{2k-1}|^2 \leq n." class="m_7396787748141865386latex"></p>
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<li style="line-height:1.6;margin-left:1em;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif">(ii) (Bieberbach conjecture) If <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%28z%29+%3D+a_1+z+%2B+a_2+z%5E2+%2B+a_3+z%5E3+%2B+%5Cdots%7D&bg=ffffff&fg=000000&s=0" alt="{f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots}" title="{f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots}" class="m_7396787748141865386latex"> is a schlicht function, then
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Ca_n%7C+%5Cleq+n.&bg=ffffff&fg=000000&s=0" alt="\displaystyle |a_n| \leq n." title="\displaystyle |a_n| \leq n." class="m_7396787748141865386latex"></p>
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It is easy to see that the Robertson conjecture for a given value of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0" alt="{n}" title="{n}" class="m_7396787748141865386latex"> implies the Bieberbach conjecture for the same value of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0" alt="{n}" title="{n}" class="m_7396787748141865386latex">. Indeed, if <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%28z%29+%3D+a_1+z+%2B+a_2+z%5E2+%2B+a_3+z%5E3+%2B+%5Cdots%7D&bg=ffffff&fg=000000&s=0" alt="{f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots}" title="{f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots}" class="m_7396787748141865386latex"> is schlicht, and <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%28z%29+%3D+b_1+z+%2B+b_3+z%5E3+%2B+b_5+z%5E5+%2B+%5Cdots%7D&bg=ffffff&fg=000000&s=0" alt="{g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots}" title="{g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots}" class="m_7396787748141865386latex"> is the odd schlicht function given by <a style="text-decoration:underline;color:#2585b2" href="https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#gsqr" target="_blank">(3)</a>, then from extracting the <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bz%5E%7B2n%7D%7D&bg=ffffff&fg=000000&s=0" alt="{z^{2n}}" title="{z^{2n}}" class="m_7396787748141865386latex"> coefficient of <a style="text-decoration:underline;color:#2585b2" href="https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#g-eq" target="_blank">(4)</a> we obtain a formula </p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em" align="center"><img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++a_n+%3D+%5Csum_%7Bj%3D1%7D%5En+b_%7B2j-1%7D+b_%7B2%28n%2B1-j%29-1%7D&bg=ffffff&fg=000000&s=0" alt="\displaystyle a_n = \sum_{j=1}^n b_{2j-1} b_{2(n+1-j)-1}" title="\displaystyle a_n = \sum_{j=1}^n b_{2j-1} b_{2(n+1-j)-1}" class="m_7396787748141865386latex"></p>
<p style="direction:ltr;font-size:14px;line-height:1.4em;color:#444;font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;margin:0 0 1em"> for the coefficients of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0" alt="{f}" title="{f}" class="m_7396787748141865386latex"> in terms of the coefficients of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%7D&bg=ffffff&fg=000000&s=0" alt="{g}" title="{g}" class="m_7396787748141865386latex">. Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0" alt="{n}" title="{n}" class="m_7396787748141865386latex"> from the Robertson conjecture for the same value of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0" alt="{n}" title="{n}" class="m_7396787748141865386latex">. We remark that <a style="text-decoration:underline;color:#2585b2" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1574687" target="_blank">Littlewood and Paley had conjectured</a> a stronger form <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%7Cb_%7B2k-1%7D%7C+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0" alt="{|b_{2k-1}| \leq 1}" title="{|b_{2k-1}| \leq 1}" class="m_7396787748141865386latex"> of Robertson's conjecture, but this was disproved for <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bk%3D3%7D&bg=ffffff&fg=000000&s=0" alt="{k=3}" title="{k=3}" class="m_7396787748141865386latex"> <a style="text-decoration:underline;color:#2585b2" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1574865" target="_blank">by Fekete and Szegö</a>.</p>
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To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7Bf%28z%29%7D%7Bz%7D%7D&bg=ffffff&fg=000000&s=0" alt="{\log \frac{f(z)}{z}}" title="{\log \frac{f(z)}{z}}" class="m_7396787748141865386latex">, known as the <em>Milin conjecture</em>. Next, one continuously enlarges the image <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%28D%280%2C1%29%29%7D&bg=ffffff&fg=000000&s=0" alt="{f(D(0,1))}" title="{f(D(0,1))}" class="m_7396787748141865386latex"> of the schlicht function to cover all of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{{\bf C}}" title="{{\bf C}}" class="m_7396787748141865386latex">; done properly, this places the schlicht function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0" alt="{f}" title="{f}" class="m_7396787748141865386latex"> as the initial function <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf+%3D+f_0%7D&bg=ffffff&fg=000000&s=0" alt="{f = f_0}" title="{f = f_0}" class="m_7396787748141865386latex"> in a sequence <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%28f_t%29_%7Bt+%5Cgeq+0%7D%7D&bg=ffffff&fg=000000&s=0" alt="{(f_t)_{t \geq 0}}" title="{(f_t)_{t \geq 0}}" class="m_7396787748141865386latex"> of univalent maps <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf_t%3A+D%280%2C1%29+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0" alt="{f_t: D(0,1) \rightarrow {\bf C}}" title="{f_t: D(0,1) \rightarrow {\bf C}}" class="m_7396787748141865386latex"> known as a <a style="text-decoration:underline;color:#2585b2" href="https://en.wikipedia.org/wiki/Loewner_differential_equation#Loewner_chain" target="_blank">Loewner chain</a>. The functions <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf_t%7D&bg=ffffff&fg=000000&s=0" alt="{f_t}" title="{f_t}" class="m_7396787748141865386latex"> obey a useful differential equation known as the <a style="text-decoration:underline;color:#2585b2" href="https://en.wikipedia.org/wiki/Loewner_differential_equation" target="_blank">Loewner equation</a>, that involves an unspecified forcing term <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%5Cmu_t%7D&bg=ffffff&fg=000000&s=0" alt="{\mu_t}" title="{\mu_t}" class="m_7396787748141865386latex"> (or <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%5Ctheta%28t%29%7D&bg=ffffff&fg=000000&s=0" alt="{\theta(t)}" title="{\theta(t)}" class="m_7396787748141865386latex">, in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bf%28z%29%7D&bg=ffffff&fg=000000&s=0" alt="{f(z)}" title="{f(z)}" class="m_7396787748141865386latex">, <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bg%28z%29%7D&bg=ffffff&fg=000000&s=0" alt="{g(z)}" title="{g(z)}" class="m_7396787748141865386latex">, or <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7Bf%28z%29%7D%7Bz%7D%7D&bg=ffffff&fg=000000&s=0" alt="{\log \frac{f(z)}{z}}" title="{\log \frac{f(z)}{z}}" class="m_7396787748141865386latex">. After some elementary calculus manipulations to ``integrate" this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of <img border="0" style="max-width:100%;height:auto;margin-bottom:12px" src="http://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0" alt="{n}" title="{n}" class="m_7396787748141865386latex">) but for which several proofs exist in the literature.
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The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the <a style="text-decoration:underline;color:#2585b2" href="https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution" target="_blank">Schramm-Loewner equation</a> (SLE), which is the focus of the next and final set of notes.
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<strong><a style="text-decoration:underline;color:#2585b2" href="https://terrytao.wordpress.com/author/teorth/" target="_blank">Terence Tao</a></strong> | 2 May, 2018 at 2:18 pm | Tags: <a style="text-decoration:underline;color:#2585b2" href="https://terrytao.wordpress.com/tag/bieberbach-conjecture/" target="_blank">Bieberbach conjecture</a>, <a style="text-decoration:underline;color:#2585b2" href="https://terrytao.wordpress.com/tag/loewner-equation/" target="_blank">Loewner equation</a>, <a style="text-decoration:underline;color:#2585b2" href="https://terrytao.wordpress.com/tag/univalent-functions/" target="_blank">Univalent functions</a>
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